Question

LetXbe a random variable whose cumulative distribution function isF(x) =P(X≤x).Themedianof a random variable is defined to be the numberx∗such thatF(x∗) =12; thatis,P(X≤x∗) =12andP(X > x∗) =12.(a) Calculate the median of an Exponential(λ) random variable.(b) IfX∼Exponential(λ), what isP(X≤E(X))?(c) There have been many psychology studies that say that people overestimate their ownintelligence. In particular, a recent one states that 65% of Americans believe that theyare “smarter than average”*. Others argue this makes no sense, and that 50% of thepopulation should be of below-average intelligence, and 50% are of above-average intel-ligence. What’s your opinion on these statements? Is it possible for 65% of people to besmarter than average?

Answer 1

(a) Calculate the median of an Exponential(λ) random variable.

m - median value

$F(m) = P(X \le m) = 1 - e^{-\lambda m} = 1/2$

From here

$m = ln(2)/\lambda$

(b) IfX∼Exponential(λ), what isP(X≤E(X))?

$P(X\le E(x)) = P(X \le 1/\lambda) = 1-e^{-\lambda * 1/ \lambda} = 1-e^{-1}=1-1/e = 0.63$

(c) My opinion that it is possible. Because the average value is not always equal to median