Question

and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have: 0 1-q Note that, in the cases of interest for us, q (0,1) 7 15 b (5 marks) Let Z-min(XY). Prove that Z ~ Geometric Show your workin Hint: It is evident that Z can only take the values 0,1,2,.... For g. {0, 1, 2, . . .), we get Z = z only if one of the three situations occur: (i) Xz and Y > z; (ii) X > z and Y -z; (iii) X-z and Y z. Another possible approach is to use the results in part (a) in or- der to calculate P(Z > z) for z E 10,1,2,...). Then show that P(Z z) = P(T

Answer 1

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