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Chapter 6: Q.6.22 (page 276)

Let W be a gamma random variable with parameters (t, 尾), 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, 尾 + n i=1 xi .

Short Answer

Expert verified

It can be seen here,(t+n,尾+鈭�i=1nxi)

Step by step solution

01

Content Introduction

Let W be a gamma random variable with parameters (t,尾).

Let W=wand X1,X2,........Xnbe are independent exponential random variable with rate w.

Show that,P(W,X1=x1,X2=x2,.......Xn=xn)is gamma with parameters(t+n,尾+鈭�i=1nx1)

02

Content Explanation

Now,

P(W=w)=f(w)=尾e-尾w(尾w)t-1蟿(t),w>0

And

P(X1=x1,W=w)=we-wxi

Now,

P(X1=x1,.....xn,W=w)=wne鈭�1n=(X1=x1,.....XN=xn鈭�W=w)P(W=w)P(W=w,X1=x1,.....,Xn=xn)=P(W=w鈭�X1=x1,....,Xn=xn)P(X1=x1,.....,Xn=xn)

03

Conclusion

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

(t+n,尾+鈭�i=1nxi

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