Question

(5) [4 pts] Let XBernoulli(p). Define Z 3x-1. (a) Is Z a random variable? Why? (b) Show that E[Z] 2p (c) Show that Elz?] 4p. (d) What is the variance of Z?

Answer 1

a) since X is random variable

Z is transformation of X

Z is also a random variable

b)

X = 1 with probability p

X = 0 with probability 1-p

Z = 3^1 -1 = 2 with probability p

Z = 3^0 -1 = 0 with probability 1-p

E(Z) = 2 * p + 0 * (1-p) = 2p

c)

E(Z^2) = 2^2 * p + 0 ^2 * (1-p) = 4p

d)

Var(Z) = E(Z^2) - (E(Z))^2

= 4p - (2p)^2

=4p -4p^2

=4p(1-p)