134 CHAPTERS DIUL 5.9 Prove Theorem 5.1. What is th quence [1/2,1/4,/ +X3, with

Question

134 CHAPTERS DIUL 5.9 Prove Theorem 5.1. What is th quence [1/2,1/4,/ +X3, with each Xi IID uniformon the outcomes k = 1, 2, 3, 4 Let S=Xi+X2 PMF of S? 5.10 5.11 What are the first four terms of the convolution of the infinite sequence u 1/16,...] with itself? 5.12 What is the convolution of the infinite sequence [1,1,1,..] with itself? If X and Yare independent random variables with means x and ), respect) variances and , respectively, what are the mean and variance of2- constants a and b? 5.13 and = aX + bY for lli random variables with Pr [Xi = 11 =pa =pand

Explanation / Answer

Since each out if equally likley so

P(Xi=k) = 1/4

And since X1, X2 and X3 are independent so

P(S=s) = P(X1=x1) P(X2=x2)P(X3=x3)

Following table shows the calculations:

Following table shows the pmg of S:

X1 X2 X3 P(X1=x1) P(X2=x2) P(X3=x3) S=X1+X2+X3 P(S=s)=P(X1=x1)*P(X1=x2)P(X3=x3) 1 1 1 0.25 0.25 0.25 3 0.015625 1 1 2 0.25 0.25 0.25 4 0.015625 1 1 3 0.25 0.25 0.25 5 0.015625 1 1 4 0.25 0.25 0.25 6 0.015625 1 2 1 0.25 0.25 0.25 4 0.015625 1 2 2 0.25 0.25 0.25 5 0.015625 1 2 3 0.25 0.25 0.25 6 0.015625 1 2 4 0.25 0.25 0.25 7 0.015625 1 3 1 0.25 0.25 0.25 5 0.015625 1 3 2 0.25 0.25 0.25 6 0.015625 1 3 3 0.25 0.25 0.25 7 0.015625 1 3 4 0.25 0.25 0.25 8 0.015625 1 4 1 0.25 0.25 0.25 6 0.015625 1 4 2 0.25 0.25 0.25 7 0.015625 1 4 3 0.25 0.25 0.25 8 0.015625 1 4 4 0.25 0.25 0.25 9 0.015625 2 1 1 0.25 0.25 0.25 4 0.015625 2 1 2 0.25 0.25 0.25 5 0.015625 2 1 3 0.25 0.25 0.25 6 0.015625 2 1 4 0.25 0.25 0.25 7 0.015625 2 2 1 0.25 0.25 0.25 5 0.015625 2 2 2 0.25 0.25 0.25 6 0.015625 2 2 3 0.25 0.25 0.25 7 0.015625 2 2 4 0.25 0.25 0.25 8 0.015625 2 3 1 0.25 0.25 0.25 6 0.015625 2 3 2 0.25 0.25 0.25 7 0.015625 2 3 3 0.25 0.25 0.25 8 0.015625 2 3 4 0.25 0.25 0.25 9 0.015625 2 4 1 0.25 0.25 0.25 7 0.015625 2 4 2 0.25 0.25 0.25 8 0.015625 2 4 3 0.25 0.25 0.25 9 0.015625 2 4 4 0.25 0.25 0.25 10 0.015625 3 1 1 0.25 0.25 0.25 5 0.015625 3 1 2 0.25 0.25 0.25 6 0.015625 3 1 3 0.25 0.25 0.25 7 0.015625 3 1 4 0.25 0.25 0.25 8 0.015625 3 2 1 0.25 0.25 0.25 6 0.015625 3 2 2 0.25 0.25 0.25 7 0.015625 3 2 3 0.25 0.25 0.25 8 0.015625 3 2 4 0.25 0.25 0.25 9 0.015625 3 3 1 0.25 0.25 0.25 7 0.015625 3 3 2 0.25 0.25 0.25 8 0.015625 3 3 3 0.25 0.25 0.25 9 0.015625 3 3 4 0.25 0.25 0.25 10 0.015625 3 4 1 0.25 0.25 0.25 8 0.015625 3 4 2 0.25 0.25 0.25 9 0.015625 3 4 3 0.25 0.25 0.25 10 0.015625 3 4 4 0.25 0.25 0.25 11 0.015625 4 1 1 0.25 0.25 0.25 6 0.015625 4 1 2 0.25 0.25 0.25 7 0.015625 4 1 3 0.25 0.25 0.25 8 0.015625 4 1 4 0.25 0.25 0.25 9 0.015625 4 2 1 0.25 0.25 0.25 7 0.015625 4 2 2 0.25 0.25 0.25 8 0.015625 4 2 3 0.25 0.25 0.25 9 0.015625 4 2 4 0.25 0.25 0.25 10 0.015625 4 3 1 0.25 0.25 0.25 8 0.015625 4 3 2 0.25 0.25 0.25 9 0.015625 4 3 3 0.25 0.25 0.25 10 0.015625 4 3 4 0.25 0.25 0.25 11 0.015625 4 4 1 0.25 0.25 0.25 9 0.015625 4 4 2 0.25 0.25 0.25 10 0.015625 4 4 3 0.25 0.25 0.25 11 0.015625 4 4 4 0.25 0.25 0.25 12 0.015625

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