In this exercise we look at a simple example of random variables Xn that have the property that their distributions converge to the distribution of a random variable X as n → ∞, while it is not true...

In this exercise we look at a simple example of random variables Xn that have the property that their distributions converge to the distribution of a random variable X as n → ∞, while it is not true that their expectations converge to the expectation of X. Let for n = 1, 2,... the random variables Xn be defined by

a. Let X be the random variable that is equal to 0 with probability 1. Show that for all a the probability mass functions pXn (a) of the Xn converge to the probability mass function pX(a) of X as n → ∞. Note that E[X]=0.

b. Show that nonetheless E[Xn] = 7 for all n.