Fall 2011 Strategic Practice 5: Section 3 (Continuous Distributions) - Question 3
Let Z ~ N(0, 1). Find E ((Z)) without using LOTUS, where is the CDF
of Z.

Solution: By Universality of the Uniform, F(X) ~ Unif(0, 1) for any continuous random variable X with CDF F. Therefore, E(Phi(Z)) = 1/2.

"Mathematics is the logic of certainty, butstatistics is the logic of uncertainty."