Choose the correct answer, and justify your response completely. Suppose = [0, 1], and let be drawn uniformly at random from the unit interval. Define the sequence of random variables Xn, n 1, on by Xn() = n I [0, 1 n2 ] () + n I [1 1 n ,1] () Here R is a fixed, finite real number, and the notation IA above denotes the indicator function of the set A. Let X be the random variable defined by X() = 0 for all [0, 1]. Which of the following is true?

(a) For = 1, the expected value of Xn is E[Xn] = 1 + 1/n.

(b) Regardless of the value of , Xn converges to X in probability.

(c) Regardless of the value of , Xn cannot converge to X with probability one because for any > 0, the set { : |Xn() X()| > } has positive probability.

(d) If = 1, then for any n > 51, we are guaranteed that |Xn() X()| < 1/50 no matter what the value of [0, 1] happens to be. 2

(e) Both (a) and (b) are true.

(f) Both (a) and (c) are true.

(g) Both (b) and (c) are true.

(h) All three of (a), (b), and (c) are true.

(i) All three of (a), (b), and (d) are true.

(j) All four of (a), (b), (c), and (d) are true.

(k) None of the above.