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.

Let be a multivariate normal random vector with mean and covariance matrixProve that the random variablehas a normal distribution with mean equal to and variance equal to .

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

Solution

The random variable can be written aswhere

Using the formula for the joint moment generating function of a linear transformation of a random vector and the fact that the mgf of a multivariate normal vector is we obtain where, in the last step, we have also used the fact that is a scalar, because is unidimensional. NowandPlugging the values just obtained into the formula for the mgf of , we getBut this is the moment generating function of a normal random variable with mean equal to and variance equal to (see the lecture entitled Normal distribution). Therefore, is a normal random variable with mean equal to and variance equal to (remember that a distribution is completely characterized by its moment generating function).

Let be a multivariate normal random vector with mean and covariance matrixUsing the joint moment generating function of , derive the cross-moment

Solution

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

Thus,

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