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STA3126 - Mathematical Statistics (1)
Homework 1
Due: April 14, 2020
1. [20 points] Let the cdf of random variable X be
F(x) =0 if x ≤ 0x2/8 if 0 ≤ x < 2 if x ≥ 2
Let the set Ck, Dk be
Ck = {x : 1/k ≤ x ≤ 2 − 1/k}
Dk = {x : 2 − 1/k < x < 2 + 1/k}, for k = 1, 2, · · ·
Derive the following events and their corresponding probabilities.
(a) limk→∞ Ck and P(X ∈ limk→∞ Ck)
(b) limk→∞ Dk and P(X ∈ limk→∞ Dk)
2. [20 points] Let W be a discrete random variable with cdf
0 otherwise,
where bwc is the largest integer less than or equal to w, and let X be a continuous random variable with cdf
0 otherwise.
(a) Find the probability mass function (pmf) of Y = W2.
(b) Find the probability density function (pdf) of Z = e−X.
3. [20 points] Prove the following statements.
(a) For a random variable X and any positive number k > 0, prove the following inequality:
kP(X > k) ≤ E(XI(X > k))
Note that
I(X > k) =1 if X > k
0 otherwise
(b) When the random variable X has its expectation as 0 or positive value, prove the following statement.
Hint: Derive the expectation, E(XI(X > k)) and obtain its limit. Then, apply the result of (a).
lim
k→∞
kP(X > k) = 0
4. [10 points] Suppose that the mgf of random variable X, denoted as MX(t), exists for −h < t < h (h > 0).
Prove the following statements.
(a) P(X ≥ a) ≤ e
−atMX(t) for all t satisfying 0 ≤ t < h
(b) P(X ≤ a) ≤ e
−atMX(t) for all t satisfying −h < t ≤ 0
5. [20 points] Suppose that the random variable X has the mgf MX(t) and its moments are as follows.
For ∀k = 1, 2, · · · ,
E(X2k−1) = 0 and E(X2k) = (2k)!2kk!.
(a) Find the moment generating function (mgf) of X. (Use the definition of mgf)
(b) Find the moment generating function (mgf) of Y = 2X − 1.
6. [10 points] Let φ(t) = log MX(t) where MX(t) is the mgf of a random variable X.
Show that φ
00 (0) = Var(X).
2