2.2 Moment Generating Functions

Find the moment generating function of the bivariate normal distribution given in Example 2.2.

If $\mathbf{Y} \sim N_{n}(\mathbf{\mu }, \mathbf{\Sigma })$, show that $Y_{i} \sim N\left(\mu_{i}, \sigma_{i i}\right)$.

Suppose that $\mathbf{Y} \sim N_{3}(\mathbf{\mu }, \mathbf{\Sigma })$, where

Find the joint distribution of $Z_{1} = Y_{1}+Y_{2}+Y_{3}$ and $Z_{2} = Y_{1}-Y_{2}$.

Given $\mathbf{Y} \sim N_{n}\left(\mathbf{\mu }, \mathbf{I}_{n}\right)$, find the joint density of $\mathbf{a}^{T} \mathbf{Y}$ and $\mathbf{b}^{T} \mathbf{Y}$, where $\mathbf{a}^{T} \mathbf{b} = 0$, and hence show that $\mathbf{a}^{T} \mathbf{Y}$ and $\mathbf{b}^{T} \mathbf{Y}$ are independent.

Let $\left(X_{i}, Y_{i}\right), i = 1,2, \ldots , n$, be a random sample from a bivariate normal distribution. Find the joint distribution of $(\bar{X}, \bar{Y})$.

If $Y_{1}$ and $Y_{2}$ are random variables such that $Y_{1}+Y_{2}$ and $Y_{1}-Y_{2}$ are independent $N(0,1)$ random variables, show that $Y_{1}$ and $Y_{2}$ have a bivariate normal distribution. Find the mean and variance matrix of $\mathbf{Y} = \left(Y_{1}, Y_{2}\right)^{T}$.

Let $X_{1}$ and $X_{2}$ have joint density

Show that $X_{1}$ and $X_{2}$ have $N(0,1)$ marginal distributions.

Suppose that $Y_{1}, Y_{2}, \ldots , Y_{n}$ are independently distributed as $N(0,1)$. Calculate the m.g.f: of the random vector

and hence show that $\bar{Y}$ is independent of $\sum_{i}\left(Y_{i}-\bar{Y}\right)^{2}$.

Let $X_{1}, X_{2}$, and $X_{3}$ be i.i.d. $N(0,1)$. Let

Show that $Y_{1}, Y_{2}$ and $Y_{3}$ are i.i.d. $N(0,1)$. (The transformation above is a special case of the so-called Helmert transformation.)