Question

#### Homework Answers

Answer #1

a) The value of Z is computed for each X and Y possibility here as:

 X Y Z 1 1 2 1 2 3 1 3 4 1 4 5 2 1 3 2 2 4 2 3 5 2 4 6 3 1 4 3 2 5 3 3 6 3 4 7 4 1 5 4 2 6 4 3 7 4 4 8

As each of the above outcome is equally likely and have a probability of 1/16 = 0.0625 occurrence, therefore the PDF for Z here is computed as:

 z P(z) 2 0.0625 3 0.125 4 0.1875 5 0.25 6 0.1875 7 0.125 8 0.0625

The expected value of Z now is computed here as:

Therefore 5 is the required expected value of Z here.

b) The probability here is computed using Bayes theorem as:
P(Z = 7 | Z > 5)

= P(Z = 7) / P(Z = 6, 7 or 8)

= 0.125 / (0.125 + 0.1875 + 0.0625) = 1/3

Therefore 0.3333 is the required probability here.

c) The probability that we have to play 4 games before we roll Z = 7 is computed here as:

= Probability of not getting Z = 7 in the first 4 games

= (1 - 0.125)4 = 0.5862

Therefore 0.5862 is the required probability here.

d) Probability that we get Z = 4, five times in the 10 roll is computed using binomial probability function as:

Therefore 0.0207 is the required probability here.

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