AssignmentI

$1.$In a certain city, the daily consumption of water $($in million of litres$)$ follows a

gamma distribution with $=2$ and $=3.$ If the daily capacity of that city is

$9$ million litres of water, what is the probability that on any given day the water

supply is inadequate? Find the mean and variance of the daily water consumption.

$2.$In a certain city, the daily consumption of electric power, in millions of kilowatt

hours, is a random variable $x$ having a gamma distribution with mean $=6$ and

variance ${}^2=12.$

$($a$).$ Find the values of $$ and $.$

$($b$).$ Find the probability that on any given day the daily power assumption will

exceed $12$ million kilowatt hours.

$3.$Let $x$ be a random variable with $mgfM(t).$ Prove that for $a>0$

$($a$).$ If $P(x0)=1,$

$P(xa)\frac{E(x)}{a}.$

$($b$).$ If $P(x0)=1$ and $k>0$

$P(xa)\frac{E(x^k)}{a^k}.$

$($c$).$ For $t>0,$

$P(xa)e^{-at}M(t).$

$($d$).$ For

$P(xa)e^{-at}M(t).$

Let $x$ be a random variable having mean $$ and standard deviation $$ and $mgf$

$E(\frac{x-}{})=0,E[(\frac{x-}{}{)}^2]=1$

and

$E\{exp[t(\frac{x-}{})]\}=e^{-\frac{t}{}}M(\frac{t}{}),|t|$

$5.$Let $xbeN(0,1).$ Use the moment generating functions techniques $to$ show that

$Y=x^{2}is{}^2(1).$

Hint: Evaluate the integral that represent $E(e^{t{x}^2})by$ writing