Let Φ be the standard normal distribution function, and let X be a normal random variable...

Let Φ be the standard normal distribution function, and let X be a normal random variable with mean µ and variance 1. We want to find E[Φ(X)]. To do so, let Z be a standard normal random variable that is independent of X. and let

(a) Show that E[I|X = x] = Φ (x).

---

(b) Show that E[Φ (X)] = P{Z}.

---

(c) Show that E[Φ (X] = Φ .

The preceding comes up in statistics. Suppose you are about to observe the value of a random variable X that is normally distributed with an unknown mean µ and variance 1, and suppose that you want to test the hypothesis that the mean µ is greater than or equal to 0. Clearly you would want to reject this hypothesis if X is sufficiently small. If it results that X = x, then the p-value of the hypothesis that the mean is greater than or equal to 0 is defined to be the probability that X would be as small as x if µ were equal to 0 (its smallest possible value if the hypothesis were true). (A small p-value is taken as an indication that the hypothesis is probably false.) Because X has a standard normal distribution when µ = 0, the p-value that results when X = x is Φ (x). Therefore, the preceding shows that the expected p-value that results when the true mean is Φ