Problems tagged with "maximum likelihood"

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.

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 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.

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