Question

(a) If X has a gamma distribution with parameters (t, λ), what is the distribution of cX, c > 0?

(b) Show that $\frac{{𝒳}_{2n}^2}{2𝝀}$has a gamma distribution with parameters n and λwhen nis a

positive integer and ${𝒳}_{2n}^2$is a chi-squared random variable with 2n degrees of freedom.

Short answer

(a) $cX~𝚪\left(t,\raisebox{1ex}{$𝝀$}\!\left/ \!\raisebox{-1ex}{$c$}\right.\right)$and

(b) $\frac{{𝒳}_{2n}^2}{2𝝀}~𝜞\left(n,𝜆\right)$

Step-by-step solution

Part (a) : Given information

$X~𝛤(t,𝜆)$

We need to find the distribution of$cX$

Part (a) : Calculations

We know that the M.G.F. of gamma distribution with parameter$\left(n,𝜆\right)$is

localid="1651051186085" $M_x\left(t\right)={\left(1-\frac{t}{𝜆}\right)}^{-n}=E\left(e^{tX}\right)$

Therefore we have

$$\begin{gathered} M_{cX}\left(t\right)=E\left[e^{\left(ct\right)X}\right]=M_X\left(ct\right) \\ \Rightarrow M_X\left(ct\right)={\left(1-\frac{ct}{𝜆}\right)}^{-n} \end{gathered}$$

which is nothing but the distribution function of gamma distribution with parameters $\left(t,\raisebox{1ex}{$𝝀$}\!\left/ \!\raisebox{-1ex}{$c$}\right.\right)$, i.e.

$cX~𝚪\left(t,\raisebox{1ex}{$𝝀$}\!\left/ \!\raisebox{-1ex}{$c$}\right.\right)$

Part (b) : Given information

Here we need to show thatis a gamma distribution with parameters n andλ