Mathematical Statistics
Fall 2024
Midterm 1
1. [5pts] Suppose X is a random variable with mean µ and variance σ
2 ≠ 0. Which of the following variables always has mean 0 and variance 1? Circle none, one, or more.
(a) Z = σ2/X − µ
(b) Z = Xσ2 + µ
(c) Z =
σ/µ−X
(d) Z = Xσ + µ
(e) Z =
σ2/X−µ
2. [5pts] Suppose X and Y are independent Bernoulli(0.5) random variables. Let Z1 = 2X −1, Z2 = 2Y − 1, Z3 = Z1Z2. Are Z1, Z2, and Z3 independent? Explain your answer.
3. Let X1, X2, X3, . . . be random variables with mean 0. Assume that
Define Sn = Σn
j=1 Xj .
(a) [5pts] Compute Var [Sn] in terms of n. Your answer should not contain any series.
(b) [5pts] Let an be any sequence of (non random) numbers with limn→∞|an| = ∞. For any > 0 use your answer to part (a) to upper-bound P(|Sn/an| > ). Does Sn/an converge to zero in probability?
4. The mean of a random integer chosen uniformly from {1, 2, 3, . . . , n} is (n + 1)/2 and the variance is (n
2 − 1)/12. Suppose the random variable X is uniformly chosen from the set {1, 2, . . . , N} where N is a Geometric(1/2) random variable. Recall that E [N] = 2 and Var [N] = 2.
(a) Compute E [X].
(b) Compute Var [X].