Question

(6) Each morning, twenty five police cadets are arranged like the entries of a 5 x 5 matrix. Each cadet forgets their cap with probability 1 -p, independently of the other cadets. Define random variables or more cadets in row i forgot their cap one 0:otherwise and or more cadets in column j forgot their cap : one Y,= 0 : otherwise X1 + X2++X +Yi+Y2++Y, Lastly, let Z (a) What is E(X,)? (b) What is P(X3 1 and Y21)? (c) Determine E(Z). (d) Determine the variance of Z.

Answer 1

(a) P(X1 1)1 - P(X1 0) 1 -p E(X1) (b) P(X3 1 and Y2 1) 1 P(X3 (Using P(An B) 1 - P(A U B)) - 1 - P(X3 0) P(Y2 0)P(X3 0, Y3 0) - 1 -2p 5 P(X1 1 1 - p 0 or Y2 0) 5 E(Y)10(1 - p) (since E(X;) E(Y) E(X (c) E(Z) _ i-1 i-1 5 (d) Var(Z) Var (X) Var(Y)2 Cov(Xi, Y) (1) i<j -1 independent and Yi, Y2,, Y (since X1, X2,., X5 Var (X) E(X) - E (X) P(X 1) (1- p) = (1 - p)p Similarly, Var(Y (1-p)p Cov(Xi, Y) E(xY) E(X)E(Y) = P(Xi 1,Y 1) (1-p)2 1 - 2p5 -p-12p5 -p10 (since P(Xi 1,Y Then from (1) Var (Z) 10p (1 - p)-20p (1 -p)10p5 10p3(110p3 20p) independent) are are -p9 (1 - p) (b)) 1) 1 2p3- p and can be computed as - 10p10 20p20p10 5