Question

9. (5 marks) Consider a Gamma random variable, Y ~ Ganzma(α = n/2, β). Find the moment- generating function of U = c Y. If U ~ , what is c?

Answer 1

9)

Here $Y\sim Ga(\alpha=\frac{n}{2},\beta)$

So, MGF of Y = $M_{Y}(t)=(1-\beta t)^{-\frac{n}{2}}$

Now, U=cY, so MGF of U is $M_{U}(t)=E(e^{tU})=E[e^{t[cY]}]=M_{Y}(ct)=(1-\beta ct)^{-\frac{n}{2}}$ --------(1)

Also, we know the if $U\sim \chi _{n}^{2}$ , then $M_{U}(t)=(1-2t)^{-\frac{n}{2}}$ ----------(2)

On comparing (1) and (2), we get, $\beta c=2\Rightarrow c=\frac{1}{\beta}$