Problems tagged with "Gaussians"

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.

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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 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.