Q. 6.7 - A First Course in Probability

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 λ when n is a

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

Step-by-Step Solution

Answer

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

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

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

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

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 and λ