Question

Let X = 1 if a randomly selected vehicle passes an emissions test and X = 0 otherwise. Then X is a Bernoulli rv with pmf p(1) = p and p(0) = 1 - P. (a) Compute E(x2). E(x²) = (b) Show that V(X) = p(1-P). = e(O )-tecon? = P(1-P) (c) Compute Exº). E(X) =

Answer 1

The probability mass function of X is

$P(X =x) = p^x*(1-p)^{1-x},x=0,1$

a) $E(X^2) = 1*P(X =1)+0*P(X = 0) = (1*p) + (0*1-p) = p$

b) $V(X) = E(X^2) - [E(X)]^2 = p - p^2 = p(1-p)$

c) $E(X^{85}) = 1^{85}*P(X =1)+0^{85}*P(X = 0) = (1*p) + (0*1-p) = p$