$7\mathrm{.}6.$Let $Z_n$ be ${}^2(n).$ Then the $mgf$ of $Z_n$ is Then mean and variance

of $n$ and $2n$ respectively. Let $Y_n=\frac{Z_n-n}{\sqrt[\phantom{2}]{2n}}.$

$($a$).$ Find the $mgf{M}_n(t)$ of $Y_n$

$($b$).$ Compute the limit $\underset{n}{lim}{M}_n(t).$

Let $x_n$ be a gamma distribution with parameter $=n$ and $,$ where $$ is not a

function of $n.$ Let $Y_n=\frac{x_n}{n}.$

$($a$).$ Find the $mgf{M}_n(t)$ of $Y_n$

$($b$).$ Compute the limit $\underset{n}{lim}{M}_n(t).$

Let $Z_n$ be ${}^2(n)$ and let $W_n=\frac{Z_n}{n^2}.$

$($a$).$ Find the mgf $M_n(t)$ of $W_n$

$($b$).$ Compute the limit $\underset{n}{lim}{M}_n(t).$

Let $Z_n$ be a Poisson distribution with parameter $=n.$ Let a random variable

$Y_n=\frac{Z_n-n}{\sqrt[\phantom{2}]{n}}.$

$($a$).$ Find the $mgf{M}_n(t)$ of $Y_n$

$($b$).$ Compute the limit $\underset{n}{lim}{M}_n(t).$

Let $x_n$ be a binomial random with parameters $n$ and $p.$

$($a$).$ Find the mgf of $x_n.$

Let

$Z_n=\frac{x_n-np}{\sqrt[\phantom{2}]{npq}}.$

$($b$).$ Find the mgf $M_{Z_n}(t)$ of $Z_n.$

$($c$).$ Compute the limit $\underset{n}{lim}{M}_{Z_n}(t).$

Let $x$ be a Poisson distribution with mean $.$

$($a$).$ Let

$Z=\frac{x-}{\sqrt[\phantom{2}]{}}.$

Find the $mgf{M}_Z(t)$ of $Z.$

$($b$).$ Compute the limit $\underset{}{lim}{M}_Z(t)\mathrm{.}6\mathrm{.}6$