Stats 510
Homework 2
Issued September 19, 2024, due by 11:59pm September 26, 2024
1. Do problems 1.47, 1.53, 2.2, 2.4., 2.8, 2.13.
2. Let X, Y, Z be real-valued continuous random variables with the pdf, respectively, fX(x) = √
1
2π
e
−x
2/2
, fY (y) = √
1
8π
e
−y
2/8
, and fZ(z) = √
1
8π
e
−(z−1)2/8
.
(i) Show directly that the random variables X , Y/2, (Z − 1)/2 and (1 − Z)/2 all have identical distributions. (Hint: Examine the corresponding cdf’s.).
(ii) Show that P(X > 0) = 1/2 and in fact, P(X > 0) = P(Y ≥ 0) = P(Z ≤ 1).
(iii) Let U be a chi squared random variable with 1 degree of freedom. Show that P(U ≤ 1) < P(X ≤ 1).
3. This question asks you to prove a theorem in the lecture notes (Theorem 2 in ”Probability integral transform” of Section 2.1). Let X be a real-valued random variable with cdf FX (x). Recall that the
inverse function for the (right-continuous) FX can be defined as follows, for 0 < y < 1,
FX(−)1 (y) := min{x : FX (x) ≥ y}.
Let U be uniform. random variable in (0, 1), and define Z = FX(−)1 (U). Show that Z has the same
distribution as that of X in the following two scenarios:
(i) X is a discrete random variable taking values in a finite set X = {a1,..., ak } ⊂ R, for some k ∈ N. (ii) X is a continuous random variable.