Problem

A discrete random variable $X$ takes values $r=0,1,2$ with probabilities $P(X=r)$ as given in the following table.

$r$	0	1	2
$P(X=r)$	$a$	$2a$	$b$

(a) Write down the probability generating function of $X$, and use it to find an expression for $\mathrm{E}(X)$ in terms of $a$ and $b$.

(b) Show that $\operatorname{Var}(X)=2b+2(a+b)(1-2a-2b)$.

The random variable $Y$ is defined by $Y=X_1+X_2+X_3+\cdots+X_{10}$ where $X_1,X_2,X_3,\ldots,X_{10}$ are ten independent observations of $X$.

(c) Using the probability generating function of $Y$, and your answer to part (a), show that $\mathrm{E}(Y)=10\mathrm{E}(X)$.

(d) For the case $b=0$, define fully the distribution of $Y$.