Problem 3.8. A natural first guess to how one might define convergence in distribution would be to say that for a sequence of random variables {Xn} and a random variable X, the sequence {Xn} converges in distribution to X if P(Xn E A) - P(X E A), for all events A C R. (1) For the purpose of this question, if (1) holds for all events A C R, we will say that {Xn} converges setwise to X. However, as we saw in class, this is not the definition of convergence in dis- tribution. This question will illustrate why the proposed definition given in (1) is "incorrect". That is to say, this question will show that, even in very simple situations, the notion of setwise convergence is too restrictive. Recall that for random variables {Yn} and Y, we say that Yn > Y as n -> co if for each y E {y : Fy is continuous at y}, we have that P( Yn oo, where Fy denotes the cumulative distribution function of Y. Consider a sequence of random variables {Xn} such that P (Xn = ? ) = 1 for each n E N and a random variable X such that P(X = 0) = 1. (a) (2 points) Denote the cumulative distribution function of X by Fx, and let Cx = {x : Fx is continuous at x}. Describe the set Cx by specifying all of the real numbers that belong to Cx. (b) (4 points) Using the characterization of convergence in distribution given in (2), show that Xn X. (c) (4 points) Show that Xn does not converge setwise to X. That is, show that there is some event A C R such that the convergence in (1) fails to hold