1. Law of Total Expectation The so-called Wald's Identity states the following: Let X1, X2,….. be a sequence of independent and identically distributed random variables with finite mean. Let further N be a nonnegative integer-valued random variable independent of the X1, X2,... with finite mean. We define the random sum SN = X1 ++ XN. Then E[SN] = E[N] E[X1]. Prove Wald's identity by conditioning on {N = n} and by using the law of total expectation.

A random number generator selects two integers from 1 to 30. What is the probability that both numbers are less than 12?

The conditional probability of E given that F occur is P(EF)= _____________. So in rolling a die the conditional probability of the event E. “getting a six,” given that the event F, “getting an even number.” has occurred is P(EF)= ____________.