Q-6.7. - A First Course in Probability

Question

1. If X has a gamma distribution with parameters $(n,𝜆)$ what is the distribution of $cX, c>0$
2. Show that $\frac{{𝒳}_{2n}^2}{2𝜆}$ has a gamma distribution with parameters $(n,𝝀)$ 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

1. $cX~𝚪(t,𝛌/c)$
   
2. $\frac{{𝒳}_{2n}^2}{2n}~𝚪(n,𝜆)$
   

Step 1. Content Introduction.

The derivative of the CDF is the probability density function $f\left(x\right)$, abbreviated PDF if it exists. A distribution function $F_x$  describes each random variable X.

Step 2. Explanation (Part a).

We are given that X has a Gamma distribution with parameters $t and \lambda$. Let's find the CDF of cX. We have that

$$\begin{gathered} F_{cX}\left(z\right)=P (cX\le z) \\ =P(X\le \frac{z}{c})=F_X\left(\frac{z}{c}\right) \end{gathered}$$

Hence,

$f_{cX}\left(z\right)=\frac{d}{dz}{F}_X\left(\frac{z}{c}\right)=f_X\left(\frac{z}{c}\right).\frac{1}{c}$

which implies that,

$$\begin{gathered} f_{cX}\left(z\right)=\frac{{\lambda }^t}{\tau \left(t\right)}{\left(\frac{z}{c}\right)}^{t-1}{e}^{-\lambda \frac{z}{c}}.\frac{1}{c} \\ =\frac{{\left({\displaystyle \frac{\lambda }{c}}\right)}^t}{\tau \left(t\right)}{z}^{t-1}{e}^{-\frac{\lambda }{c}z} \end{gathered}$$

So, we see that $cX~𝚪 (t,\frac{\lambda }{c})$

Step 3. Explanation (Part b)

Take any z > 0 and define $Z: \frac{1}{2\lambda }{X^2}_{2n}$, we have that

$$\begin{gathered} F_z\left(z\right)=P(Z\le z) \\ =P(\frac{1}{2\lambda }{X^2}_{2n}\le z) \\ =P({X^2}_{2n}\le 2\lambda z) \\ ={F_{x^2}}_{2n}\left(2\lambda z\right) \end{gathered}$$

Hence we get that

$$\begin{gathered} f_Z\left(z\right)=\frac{d}{dz}{F}_z\left(z\right) \\ =\frac{d}{dz}{F}_{{x^2}_{2n}}\left(2\lambda z\right) \\ =\frac{{\lambda }^n}{\tau \left(n\right)}{z}^{n-1}{e}^{-\lambda z} \end{gathered}$$

A chi-squared random variable with 2n degrees of freedom can be regarded as being the sum of n independent chi-square random variables each with 2 degrees of freedom (which for Example is equivalent to an exponential random variable with a parameter ). Hence Proposition ${X^2}_{2n}$ is a gamma random variable with parameters $(n,1/2)$ and the results now follow from part (a)