Q.6.22 - A First Course in Probability

Question

Let W be a gamma random variable with parameters (t, β), and suppose that conditional on W = w, X1, X2, ... , Xn are independent exponential random variables with rate w. Show that the conditional distribution of W given that X1 = x1, X2 = x2, ... , Xn = xn is gamma with parameters  t + n, β + n i=1 xi  . 

Step-by-Step Solution

Answer

It can be seen here, $(t+n, \beta +\sum _{i=1}^n{x}_i)$

Step 1: Content Introduction

Let W be a gamma random variable with parameters $(t, \beta )$.

Let $W=w$ and $X_1, X_2,........X_n$ be are independent exponential random variable with rate w.

Show that, $P(W, X_1=x_1, X_2=x_2,.......X_n=x_n)$ is gamma with parameters $(t+n, \beta +\sum _{i=1}^n{x}_1)$

Step 2: Content Explanation

Now,

$$\begin{gathered} P(W=w)=f\left(w\right) \\ =\frac{\beta e^{-\beta w}{\left(\beta w\right)}^{t-1}}{\tau \left(t\right)}, w>0 \end{gathered}$$

And 

$P(X_1=x_1, W=w)=w{e}^{-w{x}_i}$

Now,

$$\begin{gathered} P(X_1=x_1,.....x_n, W=w)=w^n{e}^{\sum _1^n} \\ =\frac{(X_1=x_1,.....X_N=x_n\cap W=w)}{P(W=w)} \\ P(W=w,X_1=x_1,.....,X_n=x_n)=\frac{P(W=w\cap X_1=x_1,....,X_n=x_n)}{P(X_1=x_1,.....,X_n=x_n)} \end{gathered}$$

Step 3: Conclusion

Therefore it can be seen that this is a form of gamma distribution with parameters

$(t+n, \beta +\sum _{i=1}^n{x}_i$