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Multivariate normal distribution - Exercise set 1

This exercise set contains some solved exercises on the multivariate normal distribution. The theory needed to solve these exercises is introduced in the lecture entitled Multivariate normal distribution.

Exercise 1.1

Let [eq1] be a multivariate normal random vector with mean [eq2]and covariance matrix[eq3]Prove that the random variable[eq4]has a normal distribution with mean equal to $3$ and variance equal to $7$.

Hint: use the joint moment generating function of X and its properties.

Solution

The random variable Y can be written as[eq5]where [eq6]

Using the formula for the joint moment generating function of a linear transformation of a random vector [eq7]and the fact that the mgf of a multivariate normal vector X is [eq8]we obtain [eq9]where, in the last step, we have also used the fact that $t$ is a scalar, because Y is unidimensional. Now[eq10]and[eq11]Plugging the values just obtained into the formula for the mgf of Y, we get[eq12]But this is the moment generating function of a normal random variable with mean equal to $3$ and variance equal to $7$ (see the lecture entitled Normal distribution). Therefore, Y is a normal random variable with mean equal to $3$ and variance equal to $7$ (remember that a distribution is completely characterized by its moment generating function).

Exercise 1.2

Let [eq1] be a multivariate normal random vector with mean [eq14]and covariance matrix[eq15]Using the joint moment generating function of X, derive the cross-moment[eq16]

Solution

The joint mgf of X is [eq17]The third-order cross-moment we want to compute is equal to a third partial derivative of the mgf, evaluated at zero:[eq18]The partial derivatives are[eq19]

Thus,[eq20]

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