Question

7.158 Let G . Deicrhm Call this number of performed until the first success occurs. 7.159 Bernoulli trials are trials X.Next, X more Bernoulli trials are this second group of Bernoulli trials. a) Obtain E(Y) by using a heuristic argument. b) Obtain E(Y) by using a formal mathematical argument. Hint: Condition on X. c) Determine Var(Y) performed. Let Y be the number of successes in 7160 Let 2 he a nositive real number Sunnose that for each n A the random variabla V

Answer 1

(a) Observe

Y is the number of success in X number of independent bernoulli trials. So if the probability of success is p then the avegare number of success in X independent bernoulli trials will be Xp. Then again X is the number of trials performed before a success occurs. Hence the average number of trials needed to get a success is 1/p and X is independent of Y so E(Y) = 1/p * p = 1

(b) E(Y)=E(E(Y|X)) = E(X*p) = 1/p * p = 1

(c) E(Y^2) = E(E(Y^2|X)) = E( X(X-1)(p^2)+1) =(p^2) E(X^2-X)+1 = (p^2)*(2-p)/(p^3) +1= 2-p/p +1 =2/p

Var(Y) = 2/p - 1= (2-p)/p