All Tags

Consider the data set shown below:

What is the predicted value of \(y\) at \(x = 3\) if the 3-nearest neighbor rule is used?

Suppose a linear prediction rule \(H(\vec x; \vec w) = \Aug(\vec x) \cdot\vec w\) is parameterized by the weight vector \(\vec w = (3, -2, 5, 2)^T\). Let \(\vec z = (1, -1, -2)^T\). What is \(H(\vec z)\)?

Suppose a line of the form \(H(x) = w_0 + w_1 x\) is fit to a data set of points \(\{(x_i, y_i)\}\) in \(\mathbb R^2\) by minimizing the mean squared error. Let the mean squared error of this predictor with respect to this data set be \(E_1\).

Next, create a new data set by adding a single new point to the original data set with the property that the new point lies exactly on the line \(H(x) = w_0 + w_1 x\) that was fit above. Let the mean squared error of \(H\) on this new data set be \(E_2\).

Which of the following is true?

Suppose a linear predictor \(H_1\) is fit on a data set \(X = \{\nvec{x}{i}, y_i\}\) of \(n\) points by minimizing the mean squared error, where each \(\nvec{x}{i}\in\mathbb R^d\).

Let \(Z = \{\nvec{z}{i}, y_i\}\) be the data set obtained from the original by standardizing each feature. That is, if a matrix were created with the \(i\)th row being \(\nvec{z}{i}\), then the mean of each column would be zero, and the variance would be one.

Suppose a linear predictor \(H_2\) is fit on this standardized data by minimizing the mean squared error.

True or False: \(H_1(\nvec{x}{i}) = H_2(\nvec{z}{i})\) for each \(i = 1, \ldots, n\).

Let \(\nvec{x}{1} = (-1, -1)^T\), \(\nvec{x}{2} = (1, 1)^T\), and \(\nvec{x}{3} = (-1, 1)^T\). Suppose \(H\) is a linear prediction function, and that \(H(\nvec{x}{1}) = 2\) while \(H(\nvec{x}{2}) = -2\) and \(H(\nvec{x}{3}) = 0\).

Let \(\nvec{x}{4} = (1,-1)^T\). What is \(H(\nvec{x}{4})\)?

Consider the data set shown below. The ``\(\times\)'' points have label +1, while the ``o'' points have label -1. Shown are the places where a linear prediction function \(H\) is equal to zero, 1, and -1.

For each of the below subproblems, calculate the total loss with respect to the given loss function. That is, you should calculate \(\sum_{i=1}^n L(\nvec{x}{i}, y_i, \vec w)\) using the appropriate loss function in place of \(L\). Note that we have most often calculated the mean loss, but here we calculate the total so that we encounter fewer fractions.

Consider the function \(f(x,y) = x^2 + |y|\). Plots of this function's surface and contours are shown below.

Which of the following are subgradients of \(f\) at the point \((0, 0)\)? Check all that apply.

Suppose gradient descent is to be used to train a perceptron classifier \(H(\vec x; \vec w)\) on a data set of \(n\) points, \(\{\nvec{x}{i}, y_i \}\). Recall that each iteration of gradient descent takes a step in the opposite direction of the ``gradient''.

Which gradient is being referred to here?

Let \(\{\nvec{x}{i}\}\) be a set of \(n\) vectors in \(\mathbb R^d\). Consider the function \(f(\vec w) = \sum_{i=1}^n \vec w \cdot\nvec{x}{i}\), where \(\vec w \in\mathbb R^d\).

True or False: \(f\) is convex as a function of \(\vec w\).

Let \(X = \{\nvec{x}{i}, y_i\}\) be a data set of \(n\) points where each \(\nvec{x}{i}\in\mathbb R^d\).

Let \(Z = \{\nvec{z}{i}, y_i\}\) be the data set obtained from the original by standardizing each feature. That is, if a matrix were created with the \(i\)th row being \(\nvec{z}{i}\), then the mean of each column would be zero, and the variance would be one.

Suppose that \(X\) and \(Z\) are both linearly-separable. Suppose Hard-SVMs \(H_1\) and \(H_2\) are trained on \(X\) and \(Z\), respectively.

True or False: \(H_1(\nvec{x}{i}) = H_2(\nvec{z}{i})\) for each \(i = 1, \ldots, n\).

Suppose a data set \(\{\nvec{x}{i}, y_i\}\) is linearly-separable.

True or false: a least squares classifier trained on this data set is guaranteed to achieve a training error of zero.

The image below shows a linear prediction function \(H\) along with a data set; the ``\(\times\)'' points have label +1 while the ``\(\circ\)'' points have label -1. Also shown are the places where the output of \(H\) is 0, 1, and -1.

True or False: \(H\) could have been learned by training a Hard-SVM on this data set.

Let \(\nvec{x}{1} = (1, 2, 0)^T\), \(\nvec{x}{2} = (-1, -1, -1)^T\), \(\nvec{x}{3} = (2, 2, 0)^T\), \(\nvec{x}{4} = (0, 2, 0)\).

Suppose a prediction function \(H(\vec x)\) is learned using kernel ridge regression on the above data set using the kernel \(\kappa(\vec x, \vec x') = (1 + \vec x \cdot\vec x')^2\) and regularization parameter \(\lambda = 3\). Suppose that \(\vec\alpha = (1, 0, -1, 2)^T\) is the solution of the dual problem.

Let \(\vec x = (0, 1, 0)^T\) be a new point. What is \(H(\vec x)\)?

Let \(R(\vec w)\) be the unregularized empirical risk with respect to the square loss (that is, the mean squared error) on a data set.

The image below shows the contours of \(R(\vec w)\). The dashed lines show places where \(\|\vec w\|_2\) is 2, 4, 6, etc.

Let \(\{\nvec{x}{i}, y_i\}\) be a data set of \(n\) points, with each \(\nvec{x}{i}\in\mathbb R^d\). Recall that the solution to the kernel ridge regression problem is \(\vec\alpha = (K + n \lambda I)^{-1}\vec y\), where \(K\) is the kernel matrix, \(I\) is the identity matrix, \(\lambda > 0\) is a regularization parameter, and \(\vec y = (y_1, \ldots, y_n)^T\).

Suppose kernel ridge regression is performed with a kernel \(\kappa\) that is a kernel for a feature map \(\vec\phi : \mathbb R^d \to\mathbb R^k\).

What is the size of the kernel matrix, \(K\)?

Let \(f(\vec w) = \vec a \cdot\vec w + \lambda\|\vec w \|^2\), where \(\vec w \in\mathbb R^d\), \(\vec a \in\mathbb R^d\), and \(\lambda > 0\).

What is the minimizer of \(f\)? State your answer in terms of \(\vec a\) and \(\lambda\).

Consider the data set of diamonds and circles shown below. Suppose a \(k\)-nn classifier is used to predict the label of the new point marked by \(\times\), with \(k = 3\). What will be the prediction? You may assume that the Euclidean distance is used.

Suppose a linear prediction rule \(H(\vec x; \vec w) = \Aug(\vec x) \cdot\vec w\) is parameterized by the weight vector \(\vec w = (2, 1, -3, 4)^T\). Let \(\vec z = (3, -2, 1)^T\). What is \(H(\vec z)\)?

In the following, let \(\mathcal D_1\) be a set of points \((x_1, y_1), \ldots, (x_n, y_n)\) in \(\mathbb R^2\). Suppose a straight line \(H_1(x) = a_1 x + a_0\) is fit to this data set by minimizing the mean squared error, and let \(R_1\) the be mean squared error of this line.

Now create a second data set, \(\mathcal D_2\), by doubling the \(x\)-coordinate of each of the original points, but leaving the \(y\)-coordinate unchanged. That is, \(\mathcal D_2\) consists of points \((2x_1, y_1), \ldots, (2x_n, y_n)\). Suppose a straight line \(H_2(x) = b_1 x + b_0\) is fit to this data set by minimizing the mean squared error, and let \(R_2\) be the mean squared error of this line.

You may assume that all of the \(x_i\) are unique, as are all of the \(y_i\).

Let \(\mathcal D\) be a data set of points in \(\mathbb R^d\). Suppose a linear prediction rule \(H_1(\vec x)\) is fit to the data by minimizing the risk with respect to the square loss, and suppose another linear prediction rule \(H_2(\vec x)\) is fit to the data by minimizing the risk with respect to the absolute loss.

True or False: \(H_1\) and \(H_2\) are guaranteed to be the same; that is, \(H_1(\vec x) = H_2(\vec x)\) for all inputs \(\vec x\).

Let \(\nvec{x}{1} = (-1, -1)^T\), \(\nvec{x}{2} = (1, 1)^T\), and \(\nvec{x}{3} = (-1,1)^T\). Suppose \(H\) is a linear prediction function, and that \(H(\nvec{x}{1}) = 2\), \(H(\nvec{x}{2}) = -2\), and \(H(\nvec{x}{3}) = 2\).

Let \(\vec z = (2,-1)^T\). What is \(H(\vec z)\)?

Consider the data set shown below. The ``\(\times\)'' points have label -1, while the ``\(\circ\)'' points have label +1. Shown are the places where a linear prediction function \(H\) is equal to zero (the thick solid line), 1, and -1 (the dotted lines). Each cell of the grid is 1 unit by 1 unit.

For each of the below subproblems, calculate the total loss with respect to the given loss function. That is, you should calculate \(\sum_{i=1}^n L(\nvec{x}{i}, y_i, \vec w)\) using the appropriate loss function in place of \(L\). Note that we have most often calculated the mean loss, but here we calculate the total so that we encounter fewer fractions.

Consider the function \(f(x,y) = |x| + y^4\). Plots of this function's surface and contours are shown below.

Write a valid subgradient of \(f\) at the point \((0, 1)\).

There are many possibilities, but you need only write one. For the simplicity of grading, please pick a subgradient whose coordinates are both integers, and write your answer in the form \((a, b)\), where \(a\) and \(b\) are numbers.

Consider the function \(f(x) = x^4 - x^2\) True or False: \(f\) is convex as a function of \(x\).

Suppose a data set \(\mathcal D\) is linearly separable. True or False: a Hard-SVM trained of \(\mathcal D\) is guaranteed to achieve 100\% training accuracy.

Consider the function \(f(x, y) = x^2 + xy + y^2\). Suppose that a single iteration of gradient descent is run on this function with a starting location of \((1, 2)^T\) and a learning rate of \(\eta = 1/10\). What will be the \(x\) and \(y\) coordinates after this iteration?

The image below shows a linear prediction function \(H\) along with a data set; the ``\(\times\)'' points have label +1 while the ``\(\circ\)'' points have label -1. Also shown are the places where the output of \(H\) is 0, 1, and -1.

True or False: \(H\) could have been learned by training a Hard-SVM on this data set.

Consider the data set: \(\nvec{x}{1} = (1, 0, 2)^T\)\(\nvec{x}{2} = (-1, 0, -1)^T\)\(\nvec{x}{3} = (1, 2, 1)^T\)\(\nvec{x}{4} = (1, 1, 0)^T\) Suppose a prediction function \(H(\vec x)\) is learned using kernel ridge regression on the above data set using the kernel \(\kappa(\vec x, \vec x') = (1 + \vec x \cdot\vec x')^2\) and regularization parameter \(\lambda = 3\). Suppose that \(\vec\alpha = (-1, -2, 0, 2)^T\) is the solution of the dual problem.

Let \(R(\vec w)\) be an unregularized empirical risk function with respect to some data set.

The image below shows the contours of \(R(\vec w)\). The dashed lines show places where \(\|\vec w\|_2\) is 1, 2, 3, etc.

Shown below are two conditional densities, \(p_1(x \,|\, Y = 1)\) and \(p_0(x \given Y = 0)\), describing the distribution of a continuous random variable \(X\) for two classes: \(Y = 0\)(the solid line) and \(Y = 1\)(the dashed line). You may assume that both densities are piecewise constant.

Suppose the Bayes classifier achieves an error rate of 15\% on a particular data distribution. True or False: It is impossible for any classifier trained on data drawn from this distribution to achieve better than 85\% accuracy on a finite test set that is drawn from this distribution.

Consider the data set of ten points shown below:

Suppose this data is used to build a histogram density estimator, \(f\), with bins: \([0,2), [2, 6), [6, 10)\). Note that the bins are not evenly sized.

Consider this data set of points \(x\) from two classes \(Y = 1\) and \(Y = 0\).

Suppose a histogram estimator with bins \([0,1)\), \([1, 2)\), \([2, 3)\) is used to estimate the densities \(p_1(x \given Y = 1)\) and \(p_0(x \given Y = 0)\), and these estimates are used in the Bayes classifier to make a prediction.

What will be the predicted class of a new point, \(x = 2.2\)?

Suppose a density estimate \(f : \mathbb R^3 \to\mathbb R^1\) is made using histogram estimators with bins having a length of 2 units, a width of 3 units, and a height of 1 unit.

What is the largest value that \(f(\vec x)\) can possibly have?

1/6 Show Answer

Suppose a discrete random variable \(X\) takes on values of either 0 or 1 and has the distribution:

where \(\theta\in[0, 1]\) is a parameter.

Given a data set \(x_1, \ldots, x_n\), what is the maximum likelihood estimate for the parameter \(\theta\)? Show your work.

Consider a data set of \(n\) points in \(\mathbb R^d\), \(\nvec{x}{1}, \ldots, \nvec{x}{n}\). Suppose the data are standardized, creating a set of new points \(\nvec{z}{1}, \ldots, \nvec{z}{n}\). That is, if the new points are stacked into an \(n \times d\) matrix, \(Z\), the mean and variance of each column of \(Z\) would be zero and one, respectively.

True or False: the covariance matrix of the standardized data must be the \(d\times d\) identity matrix; that is, the \(d \times d\) matrix with ones along the diagonal and zeros off the diagonal.

Suppose data points \(\nvec{x}{1}, \ldots, \nvec{x}{n}\) are drawn from an arbitrary, unknown distribution with density \(f\).

True or False: it is guaranteed that, given enough data (that is, \(n\) large enough), a Gaussian fit to the data using the method of maximum likelihood will approximate the true underlying density \(f\) arbitrarily closely.

Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\mu\) and the covariance matrix that must be estimated as its own parameter.

8 Show Answer

Suppose a data set consists of the following three measurements for each Saturday last year: \(X_1\): The day's high temperature \(X_2\): The number of people at Pacific Beach on that day \(X_3\): The number of people wearing coats on that day

Suppose the covariance between these features is calculated and placed into a \(3 \times 3\) sample covariance matrix, \(C\). Which of the below options most likely shows the sign of each entry of the sample covariance matrix?

Suppose we have two data sets, \(\mathcal{D}_1\) and \(\mathcal{D}_2\), each containing \(n/2\) points in \(\mathbb R^d\). Let \(\nvec{\mu}{1}\) and \(C^{(1)}\) be the mean and sample covariance matrix of \(\mathcal{D}_1\), and let \(\nvec{\mu}{2}\) and \(C^{(2)}\) be the mean and sample covariance matrix of \(\mathcal{D}_2\).

Suppose the two data sets are combined into a single data set \(\mathcal D\) containing \(n\) points.

Suppose a random vector \(\vec X = (X_1, X_2)\) has a multivariate Gaussian distribution. Suppose it is known that known that \(X_1\) and \(X_2\) are independent.

Let \(C\) be the Gaussian distribution's covariance matrix.

Consider the below data set which collects information on 10 pets:

Suppose a new pet is friendly and sheds fur. What does a Na\"ive Bayes classifier predict for its species?

Suppose a data set of 100 points in 10 dimensions is used in a binary classification task; that is, each point is labeled as either a 1 or a 0.

If Gaussian Naive Bayes is trained on this data, how many univariate Gaussians will be fit?

20 Show Answer

Recall that a deck of 52 cards contains:

Hearts: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Diamonds: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Clubs: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Spades: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Also recall that Hearts and Diamonds are red, while Clubs and Spades are black.

Consider this data set of points \(x\) from two classes \(Y = 1\) and \(Y = 0\).

Suppose a histogram estimator with bins \([0,2)\), \([2, 4)\), \([4, 6)\) is used to estimate the densities \(p_1(x \given Y = 1)\) and \(p_0(x \given Y = 0)\).

What will be the predicted class-conditional density for class 0 at a new point, \(x = 2.2\)? That is, what is the estimated \(p_0(2.2 \given Y = 0)\)?

Suppose \(\mathcal D\) is a data set of 100 points. Suppose a density estimate \(f : \mathbb R^3 \to\mathbb R^1\) is constructed from \(\mathcal D\) using histogram estimators with bins having a length of 2 units, a width of 2 units, and a height of 2 units.

The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin?

80 Show Answer

Suppose data points \(x_1, \ldots, x_n\) are independently drawn from a univariate Gaussian distribution with unknown parameters \(\mu\) and \(\sigma\).

True or False: it is guaranteed that, given enough data (that is, \(n\) large enough), a univariate Gaussian fit to the data using the method of maximum likelihood will approximate the true underlying density arbitrarily closely.

Suppose a continuous random variable \(X\) has the density:

where \(\theta\in(0, \infty)\) is a parameter, and where \(x > 0\).

Given a data set \(x_1, \ldots, x_n\), what is the maximum likelihood estimate for the parameter \(\theta\)? Show your work.

Let \(\mathcal D\) be a set of data points in \(\mathbb R^d\), and let \(C\) be the sample covariance matrix of \(\mathcal D\). Suppose each point in the data set is shifted in the same direction and by the same amount. That is, suppose there is a vector \(\vec\delta\) such that if \(\nvec{x}{i}\in\mathcal D\), then \(\nvec{x}{i} + \vec\delta\) is in the new data set.

True or False: the sample covariance matrix of the new data set is equal to \(C\)(the sample covariance matrix of the original data set).

Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count).

8 Show Answer

Let \(f_1\) be a univariate Gaussian density with parameters \(\mu\) and \(\sigma_1\). And let \(f_2\) be a univariate Gaussian density with parameters \(\mu\) and \(\sigma_2 \neq\sigma_1\). That is, \(f_2\) is centered at the same place as \(f_1\), but with a different variance.

Consider the density \(f(x) = \frac{1}{2}(f_1(x) + f_2(x))\); the factor of \(1/2\) is a normalization factor which ensures that \(f\) integrates to one.

True or False: \(f\) must also be a Gaussian density.

Consider the data set \(\mathcal D\) shown below.

What will be the sign of the \((1,2)\) entry of the data's sample covariance matrix?

Suppose a data set of points in \(\mathbb R^2\) consists of points from two classes: Class 1 and Class 0. The mean of the points in Class 1 is \((3,0)^T\), and the mean of points in Class 0 is \((7,0)^T\). Suppose Linear Discriminant Analysis is performed using the same covariance matrix \(C = \sigma^2 I\) for both classes, where \(\sigma\) is some constant.

Suppose there were 50 points in Class 1 and 100 points in Class 0.

Consider a new point, \((5, 0)^T\), exactly halfway between the class means. What will LDA predict its label to be?

Consider the below data set which collects information on the weather on 10 days:

Suppose a new day is not sunny but is warm. What does a Na\"ive Bayes classifier predict for whether it rained?

Suppose that a deck of cards has some cards missing. Namely, both the Ace of Spades and Ace of Clubs are missing, leaving 50 cards remaining.

Hearts: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Diamonds: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Clubs: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

Spades: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

Also recall that Hearts and Diamonds are red, while Clubs and Spades are black.

Let \(\mathcal X = \{(\nvec{x}{1}, y_1), \ldots, (\nvec{x}{n}, y_n)\}\) be a labeled dataset, where \(\nvec{x}{i}\in\mathbb R^d\) is a feature vector and \(y_i \in\{-1, 1\}\) is a binary label. Let \(\vec x\) be a new point that is not in the data set. Suppose a nearest neighbor classifier is used to predict the label of \(\vec x\), and the resulting prediction is \(-1\). (You may assume that there is a unique nearest neighbor of \(\vec x\).)

Now let \(\mathcal Z\) be a new dataset obtained from \(\mathcal X\) by scaling each feature by a factor of 2. That is, \(\mathcal Z = \{(\nvec{z}{1}, y_1), \ldots, (\nvec{z}{n}, y_n)\}\), where \(\nvec{z}{i} = 2 \nvec{x}{i}\) for each \(i\). Let \(\vec z = 2 \vec x\). Suppose a nearest neighbor classifier trained on \(\mathcal Z\) is used to predict the label of \(\vec z\).

True or False: the predicted label of \(\vec z\) must also be \(-1\).

Solution

True.

This question is essentially asking: if we scale a data set by a constant factor, does the nearest neighbor of a point change? The answer is no: therefore, the predicted label cannot change.

When we multiply each feature of a point by the same constant factor, it has the effect of scaling the data set. That is, if \(\mathcal X\) is the data shown in the top of the image shown below, then \(\mathcal Z\) is the data shown in the bottom of the image (not drawn to scale):

You can see that the nearest neighbor of \(\vec x\) in \(\mathcal X\) is point #4, and the nearest neighbor of \(\vec z\) in \(\mathcal Z\) remains point #4. As such, the predicted label does not change.

Suppose a linear prediction function \(H(\vec x) = w_0 + w_1 x_1 + w_2 x_2\) is trained to perform classification, and the weight vector is found to be \(\vec w = (2, -1, 2)^T\).

The figure below shows four possible decision boundaries: \(A\), \(B\), \(C\), and \(D\). Which of them is the decision boundary of the prediction function \(H\)?

You may assume that each grid square is 1 unit on each side, and the axes are drawn to show you where the origin is.

Suppose a linear prediction function \(H(\vec x) = w_1 \phi_1(\vec x) + w_2 \phi_2(\vec x) + w_3 \phi_3(\vec x) + w_4 \phi_4(\vec x)\) is fit using the basis functions

where \(\vec x = (x_1, x_2)^T\) is a feature vector in \(\mathbb R^2\). The weight vector \(\vec w\) is found to be \(\vec w = (1, -2, 3, -4)^T\).

Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)?

3 Show Answer

Suppose a linear regression model \(H(\vec x) = w_0 + w_1 x_1 + w_2 x_2\) is trained by minimizing the mean squared error on the data shown below, where \(x_1\) and \(x_2\) are the features and \(y\) is the target:

True or False: there is a unique weight vector \(\vec w = (w_0, w_1, w_2)^T\) minimizing the mean squared error for this data set.

Suppose a linear prediction function \(H(\vec x) = w_0 + w_1 x_1 + w_2 x_2\) is fit to data and the following are observed:

At a point, \(\nvec{x}{1} = (1, 1)^T\), the value of \(H\) is 4.

At a point, \(\nvec{x}{2} = (1, 0)^T\), the value of \(H\) is 0.

At a point, \(\nvec{x}{3} = (0, 0)^T\), the value of \(H\) is -4.

What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)?

0 Show Answer

Solution

The four points make a square:

Since \(H = 0\) at \(\nvec{x}{2}\), the decision boundary must pass through that point. And since \(H(\nvec{x}{1}) = 4 = -H(\nvec{x}{3})\), the points \(\nvec{x}{1}\) and \(\nvec{x}{3}\) are equidistant from the decision boundary. Therefore, the decision boundary must pass exactly halfway between \(\nvec{x}{1}\) and \(\nvec{x}{3}\). This makes a line going through \(\nvec{x}{4}\), meaning that \(\nvec{x}{4}\) is on the decision boundary, so \(H(\nvec{x}{4}) = 0\).

Choose the option which best completes the following sentence: In least squares regression, we can fit a linear prediction function \(H\) by computing the gradient of the _________ with respect to ________ and solving.

Consider the data set shown below. The 4 points marked with ``+'' have label +1, while the 3 ``\(\circ\)'' points have label -1. Shown are the places where a linear prediction function \(H\) is equal to zero (the thick solid line), 1, and -1 (the dotted lines). Each cell of the grid is 1 unit by 1 unit. The origin is not specified; it is not necessary to know the absolute coordinates of the data to complete this problem.

For each of the below subproblems, calculate the total loss with respect to the given loss function. That is, you should calculate \(\sum_{i=1}^n L(\nvec{x}{i}, y_i, \vec w)\) using the appropriate loss function in place of \(L\). Note that we have most often calculated the mean loss, but here we calculate the total so that we encounter fewer fractions.

Consider the piecewise function:

For convenience, a plot of this function is shown below:

Note that the plot isn't intended to be precise enough to read off exact values, but it might help give you a sense of the function's behavior. In principle, you can do this question using only the piecewise definition of \(f\) given above.

Recall that a subgradient of the absolute loss is:

Suppose you are running subgradient descent to minimize the risk with respect to the absolute loss in order to train a function \(H(x) = w_0 + w_1 x\) on the following data set:

Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent?

\((1, 2)^T\) Show Answer

Let \(\mathcal X = \{(\nvec{x}{1}, y_1), \ldots, (\nvec{x}{n}, y_n)\}\) be a data set for regression, with each \(\nvec{x}{i}\in\mathbb R^d\) and each \(y_i \in\mathbb R\). Consider the following function:

True or False: \(R(\vec w)\) is a convex function of \(\vec w\).

Recall that in ridge regression, we solve the following optimization problem:

where \(\lambda > 0\) is a hyperparameter controlling the strength of regularization.

Suppose you solve the ridge regression problem with \(\lambda = 2\), and the resulting solution is the weight vector \(\vec w_\text{old}\). You then solve the ridge regression problem with \(\lambda = 4\) and find a weight vector \(\vec w_\text{new}\).

True or False: each component of the new solution, \(\vec w_\text{new}\), must be less than or equal to the corresponding component of the old solution, \(\vec w_\text{old}\).

Solution

False.

While it is true that \(\|\vec w_\text{new}\|\leq\|\vec w_\text{old}\|\), this does not imply that each component of \(\vec w_\text{new}\) is less than or equal to the corresponding component of \(\vec w_\text{old}\).

The picture to have in mind is that of the contour lines of the mean squared error (which are ovals), along with the circles representing where \(\|\vec w\| = c\) for some constant \(c\). The question asked us to consider going from \(\lambda = 2\) to \(\lambda = 4\), but to gain an intuition we can think of going from no regularization (\(\lambda = 0\)) to some regularization (\(\lambda > 0\)); this won't affect the outcome, but will make the story easier to tell.

Consider the situation shown below:

When we had no regularization, the solution was \(\vec w_\text{old}\), as marked. Suppose we add regularization, and we're told that the regularization is such that when we solve the ridge regression problem, the norm of \(\vec w_\text{new}\) will be equal to \(c\), and that the radius of the circle we've drawn is \(c\). Then the solution \(\vec w_\text{new}\) will be the point marked, since that is the point on the circle that is on the lowest contour.

Notice that the point \(\vec w_\text{new}\) is closer to the origin, and it's first component is much smaller than the first component of \(\vec w_\text{old}\). However, the second component of \(\vec w_\text{new}\) is actually larger than the second component of \(\vec w_\text{old}\).

In lecture, we saw that SVMs minimize the (regularized) risk with respect to the hinge loss. Now recall, the 0-1 loss, which was defined in lecture as:

Suppose a perceptron classifier \(H(\vec x) = \vec w \cdot\vec x\) is trained on the data shown below. The points marked with ``+'' have label +1, while the ``\(\circ\)'' points have label -1. The perceptron's decision boundary is plotted as a thick solid line.

Let \(R(\vec w)\) be the risk with respect to the perceptron loss function, and let \(\vec w^*\) be the weight vector whose corresponding decision boundary is shown above.

True or False: the gradient of \(R\) evaluated at \(\vec w^*\) is \(\vec 0\).

Consider the data set shown below. The points marked with ``+'' have label +1, while the ``\(\circ\)'' points have label -1. Shown are the places where a linear prediction function \(H\) is equal to zero (the thick solid line), 1, and -1 (the dotted lines). Each cell of the grid is 1 unit by 1 unit. The origin is not specified (it is not necssary to know the absolute coordinates of any points for this problem).

Suppose that the weight vector \(\vec w\) of the linear prediction function \(w_0 + w_1 x_1 + w_2 x_2\) shown above is \((6, 0, \frac{1}{2})^T\). This weight vector is not the solution to the Hard SVM problem.

What weight vector is the solution of the Hard SVM problem for this data set?

\((3, 0, \frac{1}{4})^T\) Show Answer

Let \(\nvec{x}{1}, \ldots, \nvec{x}{n}\) be vectors in \(\mathbb R^d\) and let \(\vec w\) be a vector in \(\mathbb R^d\). Consider the function:

What is \(\frac{d}{d \vec w} f(\vec w)\)?

\(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) Show Answer

The ``inifinity norm'' of a vector \(\vec w \in\mathbb R^d\), written \(\|\vec w\|_\infty\), is defined as:

That is, it is the maximum absolute value of any entry of \(\vec w\).

Let \(R(\vec w)\) be an unregularized risk function on a data set. The solid curves in the plot below are the contours of \(R(\vec w)\). The dashed lines show where \(\|\vec w\|_\infty\) is equal to 1, 2, 3, and so on.

Let \(\tilde R(\vec w) = R(\vec w) + \lambda\|\vec w\|_\infty^2\), with \(\lambda > 0\). The point marked \(A\) is the minimizer of the unregularized risk. Suppose that it is known that one of the other points is the minimizer of the regularized risk, \(\tilde R(\vec w)\), for some unknown \(\lambda > 0\). Which point is it?

Let \(\mathcal X = \{(\nvec{x}{1}, y_1), \ldots, (\nvec{x}{n}, y_n)\}\) be a labeled dataset, where \(\nvec{x}{i}\in\mathbb R^d\) is a feature vector and \(y_i \in\{-1, 1\}\) is a binary label. Let \(\vec x\) be a new point that is not in the data set. Suppose a nearest neighbor classifier is used to predict the label of \(\vec x\), and the resulting prediction is \(-1\). (You may assume that there is a unique nearest neighbor of \(\vec x\).)

Now let \(\mathcal Z\) be a new dataset obtained from \(\mathcal X\) by subtracting the same vector \(\vec\delta\) from each training point. That is, \(\mathcal Z = \{(\nvec{z}{1}, y_1), \ldots, (\nvec{z}{n}, y_n)\}\), where \(\nvec{z}{i} = \nvec{x}{i} - \vec\delta\) for each \(i\). Let \(\vec z = \vec x - \vec\delta\). Suppose a nearest neighbor classifier trained on \(\mathcal Z\) is used to predict the label of \(\vec z\).

True or False: the predicted label of \(\vec z\) must also be \(-1\).

Suppose a linear prediction function \(H(\vec x) = w_0 + w_1 x_1 + w_2 x_2\) is trained to perform classification, and the weight vector is found to be \(\vec w = (1,-2,1)^T\).

The figure below shows four possible decision boundaries: \(A\), \(B\), \(C\), and \(D\). Which of them is the decision boundary of the prediction function \(H\)?

You may assume that each grid square is 1 unit on each side, and the axes are drawn to show you where the origin is.

Suppose a linear prediction function \(H(\vec x) = w_1 \phi_1(\vec x) + w_2 \phi_2(\vec x) + w_3 \phi_3(\vec x) + w_4 \phi_4(\vec x)\) is fit using the basis functions

where \(\vec x = (x_1, x_2, x_3)^T\) is a feature vector in \(\mathbb R^3\). The weight vector \(\vec w\) is found to be \(\vec w = (2, 1, -2, -3)^T\).

Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)?

\(-3\) Show Answer

Consider the binary classification data set shown below consisting of six points in \(\mathbb R^2\). Each point has an associated label of either +1 or -1.

What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number.

0 Show Answer

Suppose \(f(x_1, x_2)\) is a convex function. Define the new function \(g(x) = f(x, 0)\). True or False: \(g(x)\) must be convex.

Let \(f_1(x)\) be a convex function and let \(f_2(x)\) be a non-convex function. Define the new function \(f(x) = f_1(x) + f_2(x)\). True or False: \(f(x)\) must be non-convex.

Consider the data set shown below. The \(\diamond\) points marked with ``+'' have label +1, while the \(\circ\) points marked with ``\(-\)'' have label -1. Shown are the places where a linear prediction function \(H\) is equal to zero (the thick solid line), 1, and -1 (the dotted lines). Each cell of the grid is 1 unit by 1 unit. The origin is not specified; it is not necessary to know the absolute coordinates of the data to complete this problem.

For each of the below subproblems, calculate the total loss with respect to the given loss function. That is, you should calculate \(\sum_{i=1}^n L(\nvec{x}{i}, y_i, \vec w)\) using the appropriate loss function in place of \(L\). Note that we have most often calculated the mean loss, but here we calculate the total so that we encounter fewer fractions.

Note: please double-check your calculations! We are not grading your work, so partial credit is difficult to assign on this problem.

Consider the following piecewise function that is linear in each quadrant:

For convenience, a plot of this function is shown below:

Note that the plot isn't intended to be precise enough to read off exact values, but it might help give you a sense of the function's behavior. In principle, you can do this question using only the piecewise definition of \(f\) given above.

Recall that a subgradient of the absolute loss is:

Suppose you are running subgradient descent to minimize the risk with respect to the absolute loss in order to train a function \(H(x) = w_0 + w_1 x\) on the following data set:

Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent?

\((1, 3)^T\) Show Answer

Recall that in ridge regression, we solve the following optimization problem:

where \(\lambda > 0\) is a hyperparameter controlling the strength of regularization.

Suppose you solve the ridge regression problem with \(\lambda = 2\), and the resulting solution has a mean squared error of \(10\).

Now suppose you increase the regularization strength to \(\lambda = 4\) and solve the ridge regression problem again. True or False: it is possible that the mean squared error of the new solution is less than \(10\).

By ``mean squared error,'' we mean \(\frac1n \sum_{i=1}^n (y_i - \vec w \cdot\operatorname{Aug}(\nvec{x}{i}))^2\)

Let \(\mathcal X\) be a binary classification data set \((\nvec{x}{1}, y_1), \ldots, (\nvec{x}{n}, y_n)\), where each \(\nvec{x}{i}\in\mathbb R^d\) and each \(y_i \in\{-1, 1\}\). Suppose that the data in \(\mathcal X\) is linearly separable.

Let \(\vec w_{\text{svm}}\) be the weight vector of the hard-margin support vector machine (SVM) trained on \(\mathcal X\). Let \(\vec w_{\text{lsq}}\) be the weight vector of the least squares classifier trained on \(\mathcal X\).

True or False: it must be the case that \(\vec w_{\text{svm}} = \vec w_{\text{lsq}}\).

Recall that the Soft-SVM optimization problem is given by

where \(\vec w\) is the weight vector, \(\vec\xi\) is a vector of slack variables, and \(C\) is a regularization parameter.

The two pictures below show as dashed lines the decision boundaries that result from training a Soft-SVM on the same data set using two different regularization parameters: \(C_1\) and \(C_2\).

True or False: \(C_1 < C_2\).

Consider the data set shown below. The points marked with ``+'' have label +1, while the ``\(\circ\)'' points have label -1. Shown are the places where a linear prediction function \(H\) is equal to zero (the thick solid line), 1, and -1 (the dotted lines). Each cell of the grid is 1 unit by 1 unit. The origin is not specified (it is not necssary to know the absolute coordinates of any points for this problem).

Let \(\vec w'\) be the parameter vector corresponding to prediction function \(H\) shown above. Let \(R(\vec w)\) be the risk with respect to the hinge loss on this data. True or False: the gradient of \(R\) evaluated at \(\vec w'\) is \(\vec 0\).

Let \(\vec x, \vec\mu,\) and \(\vec b\) be vectors in \(\mathbb R^n\), and let \(A\) be a symmetric \(n \times n\) matrix. Consider the function:

What is \(\frac{d}{d \vec x} f(\vec x)\)?

\(2 A \vec x + \vec b\) Show Answer

The ``\(p = \frac12\)'' norm of a vector \(\vec w \in\mathbb R^d\), written \(\|\vec w\|_{\frac12}\), is defined as:

Let \(R(\vec w)\) be an unregularized risk function on a data set. The solid curves in the plot below are the contours of \(R(\vec w)\). The dashed lines show where \(\|\vec w\|_{\frac12}\) is equal to 1, 2, 3, and so on.

Let \(\tilde R(\vec w) = R(\vec w) + \lambda\|\vec w\|_{\frac12}\), with \(\lambda > 0\). The point marked \(A\) is the minimizer of the unregularized risk. Suppose it is known that one of the other points is the minimizer of the regularized risk, \(\tilde R(\vec w)\), for some unknown \(\lambda > 0\). Which point is it?

Suppose Gaussian kernel ridge regression is used to train a model on the following data set of \((x_i, y_i)\) pairs, using a kernel width parameter of \(\gamma = 1\) and a regularization parameter of \(\lambda = 0\):

Let \(\vec\alpha\) be the solution to the dual problem. What will be the sign of \(\alpha_5\)?

Shown below are two conditional densities, \(p_1(x \,|\, Y = 1)\) and \(p_0(x \given Y = 0)\), describing the distribution of a continuous random variable \(X\) for two classes: \(Y = 0\)(the solid line) and \(Y = 1\)(the dashed line). You may assume that both densities are piecewise constant.

In this problem, consider the following labeled data set of 15 points, 5 from Class 1 and 10 from Class 0.

Suppose the class conditional densities are estimated using a histogram estimator with bins: \([0, 3), [3, 5), [5, 6),\) and \([6, 10)\). Note that the bins are not all the same width!

In all of the parts below, you may write your answer either as a decimal or as a fraction.

Suppose a prediction function \(H(\vec x)\) is trained using kernel ridge regression on the data below using regularization parameter \(\lambda = 4\) and kernel \(\kappa(\vec x, \vec x') = (1 + \vec x \cdot\vec x')^2\):

\(\nvec{x}{1} = (1, 2, 0)\), \(y_1 = 1\)\(\nvec{x}{2} = (0, 1, 2)\), \(y_2 = -1\)\(\nvec{x}{3} = (2, 0, 2)\), \(y_3 = 1\)\(\nvec{x}{4} = (1, 0, 1)\), \(y_4 = -1\)\(\nvec{x}{5} = (0, 0, 0)\), \(y_5 = 1\) Suppose the solution to the dual problem is \(\vec\alpha = (2, 0, 1, -1, 2)\).

Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)?

50 Show Answer

Recall that the density of a \(d\)-dimensional ``axis-aligned'' Gaussian (that is, a Gaussian with diagonal covariance matrix \(C\)) is given by:

Consider the marginal density \(p_1(x_1)\), which is the density of the first coordinate, \(x_1\), of a \(d\)-dimensional axis-aligned Gaussian.

True or False: \(p_1(x_1)\) must be a Gaussian density.

Consider the below data set collecting information on a set of 10 flowers.

What species does a Naive Bayes classifier predict for a flower with 3 petals and no leaves?

Suppose Justin has a dartboard at home that looks like the below:

Justin uses the dartboard to determine how long the midterm will be: if he throws a dart and it lands in the shaded region, the midterm will have 17 questions; if it lands in the unshaded region, it will have 16 questions.

Assume that Justin's dart throws are drawn from a uniform distribution on the dartboard, and that the dart always hits the board (that is, the density function is constant everywhere on the dartboard, and zero off of the dartboard). Let \(X_1\) be the horizontal component of a dart throw and \(X_2\) be the vertical component. Let \(Q\) be the number of questions on the exam; since it is chosen randomly, it is also a random number.

Suppose the underlying class-conditional densities in a binary classification problem are known to be multivariate Gaussians.

Suppose a Quadratic Discriminant Analysis (QDA) classifier using full covariance matrices for each class is trained on a data set of \(n\) points sampled from these densities.

True or False: As the size of the data set grows (that is, as \(n \to\infty\)), the training error of the QDA classifier must approach zero.

Suppose a univariate Gaussian density function \(\hat f\) is fit to a set of data using the method of maximum likelihood estimation (MLE).

True or False: \(\hat f\) must be between 0 and 1 everywhere. That is, it must be the case that for every \(x \in\mathbb R\), \(0 < \hat f(x) \leq 1\).

Suppose Quadratic Discriminant Analysis (QDA) is used to train a classifier on the following data set of \((x_i, y_i)\) pairs, where \(x_i\) is the feature and \(y_i\) is the class label:

Univariate Gaussians are used to model the class conditional densities, each with their own mean and variance.

What is the prediction of the QDA classifier at \(x = 2.25\)?

Consider Justin's rectangle density. It is a parametric density with two parameters, \(\alpha\) and \(\beta\), and pdf:

A picture of the density is shown below for convenience:

In all of the below parts, let \(\mathcal X = \{1, 2, 3, 6, 9\}\) be a data set of 5 points generated from the rectangle density.

Your answers to the below problems should all be in the form of a number. You may leave your answer as an unsimplified fraction or a decimal, if you prefer.

Let \((\nvec{x}{1}, y_1), \ldots, (\nvec{x}{n}, y_n)\) be a set of \(n\) points in a binary classification problem, where \(\nvec{x}{i}\in\mathbb{R}^2\) and \(y_i \in\{0, 1\}\).

Suppose a classifier is trained by estimating the class-conditional densities with histograms using rectangular bins and applying the Bayes classification rule.

True or False: it is always possible to achieve a 100\% training accuracy with this classifier by choosing the rectangular bins to be sufficiently small. You may assume that no two points \(\nvec{x}{i}\) and \(\nvec{x}{j}\) are identical.

Does standardizing the data possibly change the prediction made by Gaussian Naive Bayes?

More precisely, let \((\nvec{x}{1}, y_1), \ldots, (\nvec{x}{n}, y_n)\) be a set of training data for a binary classification problem, where each \(\nvec{x}{i}\in\mathbb R^d\) and each \(y_i \in\{0, 1\}\). Suppose that when a Gaussian Na\"ive Bayes classifier is trained on this data set, \(\nvec{x}{1}\) is predicted to be from Class \(0\).

Now suppose that the data set is standardized by subtracting the mean of each feature and dividing by its standard deviation. This is done for the data set as a whole, not separately for each class. Let \(\nvec{z}{1}, \ldots, \nvec{z}{n}\) be the standardized data set.

True or False: when a Gaussian Na\"ive Bayes classifier is trained on the standardized data set, it must still predict \(\nvec{z}{1}\) to be from Class \(0\).

Consider the following set of 6 data points:

In the below parts, your answers should be given as numbers. You may leave your answer as an unsimplified fraction or a decimal, if you prefer.

The picture below shows the contours of a multivariate Gaussian density function:

Which one of the following could possibly be the covariance matrix of this Gaussian?

The picture below shows the decision boundary of a binary classifier. The shaded region is where the classifier predicts for Class 1; everywhere else, the classifier predicts for Class 0.

True or False: this could be the decision boundary of a Quadratic Discriminant Analysis (QDA) classifier that models the class-conditional densities as multivariate Gaussians with full covariance matrices.

Consider the following data set of four points whose feature vectors are in \(\mathbb{R}^2\) and whose labels are in \(\{-1,1\}\):

For convenience, we've plotted the data below. Each point is labeled with either the positive class (denoted by \(\times\)) or the negative class (denoted by \(\bullet\)).

Suppose an unnamed kernel classifier \(H(\vec x) = \sum_{i=1}^n \alpha_i \kappa(\nvec{x}{x}, \vec x)\) has been trained on this data using a (spherical) Gaussian kernel and kernel width parameter \(\gamma = 1\). Suppose the solution to the dual problem is found to be \(\vec\alpha = (1, 1, 1, -3)^T\).

What class will the classifier predict for the point \((2,2)\)? For convenience, we've plotted this point on the graph above as a question mark.

Shown below are two conditional densities, \(p_1(x \,|\, Y = 1)\) and \(p_0(x \given Y = 0)\), describing the distribution of a continuous random variable \(X\) for two classes: \(Y = 0\)(the solid line) and \(Y = 1\)(the dashed line). You may assume that both densities are piecewise constant.

In this problem, consider the following labeled data set of 19 points, 7 from Class 1 and 12 from Class 0.

Suppose the class conditional densities are estimated using a histogram estimator with bins: \([0, .25), [.25, .5), [.5, 75),\) and \([.75, 1.0)\).

In all of the parts below, you may write your answer either as a decimal or as a fraction.

Suppose a prediction function \(H(\vec x)\) is trained using kernel ridge regression on the data below using regularization parameter \(\lambda = 4\) and kernel \(\kappa(\vec x, \vec x') = (1 + \vec x \cdot\vec x')^2\):

\(\nvec{x}{1} = (0, 1, 1)\), \(y_1 = 1\)\(\nvec{x}{2} = (1, 1, 1)\), \(y_2 = -1\)\(\nvec{x}{3} = (2, 2, 2)\), \(y_3 = 1\)\(\nvec{x}{4} = (1, 1, 0)\), \(y_4 = -1\)\(\nvec{x}{5} = (0, 1, 0)\), \(y_5 = 1\) Suppose the solution to the dual problem is \(\vec\alpha = (-1, 1, 0, 3, -2)\).

Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)?

37 Show Answer

Consider the following set of 6 data points:

In the below parts, your answers should be given as numbers. You may leave your answer as an unsimplified fraction or a decimal, if you prefer.

Let \(X_1\) and \(X_2\) be two independent random variables. Suppose the distribution of \(X_1\) has the Gaussian density:

while the distribution of \(X_2\) has the Gaussian density:

Which one of the following pictures shows the contours of the joint density \(p(x_1, x_2)\)(the density for the joint distribution of \(X_1\) and \(X_2\))?

Suppose it is known that the distribution of a random variable \(X\) has a univariate Gaussian density function \(f\).

True or False: \(f\) must be between 0 and 1 everywhere. That is, it must be the case that for every \(x \in\mathbb R\), \(0 < f(x) \leq 1\).

Suppose that, in a binary classification setting, the true underlying class-conditional densities \(p(\vec x \given Y=0)\) and \(p(\vec x \given Y=1)\) are known to each be multivariate Gaussians with full covariance matrices. Suppose, also, that \(\pr(Y = 1) = \pr(Y = 0) = \frac{1}{2}\).

True or False: it is possible that the Bayes error in this case is exactly zero.

Consider Justin's right-triangle density. It is a parametric density with two parameters, \(\alpha\) and \(\beta\)(with \(\beta >0\)), and pdf:

A picture of the density is shown below for convenience:

Your answers to the below problems should all be in the form of a number. You may leave your answer as an unsimplified fraction or a decimal, if you prefer.

Consider the following data set of 14 points in \(\mathbb R^2\). Each point has a label in \(\{ 1, -1 \}\). Points from Class 1 are marked with \(\times\) and points from Class -1 are marked with \(\bullet\).

Suppose the class-conditional density estimates are computed from this data using a histogram density estimator using the \(1 \times 1\) bins shown in the figure above.

If these estimates are used in place of the true class-conditional densities in the Bayes classifier, what will be the training error of the classifier? That is, what percentage of the data above will be misclassified? You may leave your answer as a fraction or a decimal.

4/14 = .2857 Show Answer

The picture below shows the contours of a multivariate Gaussian density function:

Which one of the following could possibly be the covariance matrix of this Gaussian?

The picture below shows the decision boundary of a binary classifier. The shaded region is where the classifier predicts for Class 1; everywhere else, the classifier predicts for Class 0. You can assume that the shaded region extends infinitely to the left and down.

True or False: this could be the decision boundary of a classifier that estimates each class conditional density using two Gaussians with different means but the same, shared full covariance matrix, and applies the Bayes classification rule.

Suppose that Justin has a wall at home that he has painted to look like the below:

Justin uses the wall to determine how long Redemption Midterm 02 will be: if he throws a dart at the origin and it lands in the shaded region, the redemption midterm will have 14 questions; if it lands in the unshaded region, it will have 2 questions.

Assume that Justin's dart throws are drawn from a spherical Gaussian whose mean is at the origin. Let \(X_1\) be the horizontal component of a dart throw and \(X_2\) be the vertical component. Let \(Q\) be the number of questions on the exam; since it is chosen randomly, it is also a random number. You can assume that the wall is infinitely large, and that the shaded regions extend infinitely up and to the right, and down and to the left.