Question
In a certain game, an outcome or event A can occur withprobability 0.25. Let Z denote the number of times that A occurs ifyou play this game sixty times. 1) What is the distribution of Z? 2) Compute E(Z) 3) Compure Var(Z) 4) What are the smallest and largest values that the randomvariable Z can take? 5) Compute E(Z/60) 6) Compute Var(Z/60) 7) What are the smallest and largest that the random variableZ/60 can take? In a certain game, an outcome or event A can occur withprobability 0.25. Let Z denote the number of times that A occurs ifyou play this game sixty times. 1) What is the distribution of Z? 2) Compute E(Z) 3) Compure Var(Z) 4) What are the smallest and largest values that the randomvariable Z can take? 5) Compute E(Z/60) 6) Compute Var(Z/60) 7) What are the smallest and largest that the random variableZ/60 can take?
Explanation / Answer
1) Let Z denote the number of times that A occurs if you playthis game sixty times. Z follows binomial distribution with parameters n = 60, and p= 0.25 2)Since Z is binomial, E(Z) = np = 60(0.25) = 15 3) Var(Z) = npq = 60(0.25)(1-0.25) = 11.25 4) smallest and largest values: Z=0, and Z = 60. The worstcase is that the player never wins, and the best case would be towin all the 60 games 5) E(Z/60) = (1/60) E(Z) = (1/60)(15) = 0.25 6) Var(Z/60) = (1/60)^2 Var(Z) = (1/60)^2 (11.25) =0.003125 7) Since we have, from part 4, smallest and largest values:Z=0, and Z = 60 Then, for random variable Z/60, we would have Z = 0/60 = 0 andZ = 60/60 = 1 for the smallest and the largest