statistics questions

STA347 Probability I

Assignment #2

Due: October 29, 2018 before class starts

Solve the following and hand in by due date.

1. Solve Problem 3.8.12

Problem (3.8.12). Let f(x, y) = c(x2 − y2)e−x, 0 ≤ x < ∞, −x ≤ y < x. a. Find c. b. Find the marginal densities. c. Find the conditional densities.

2. Solve Problem 3.8.73

Problem (3.8.73). If X1, . . . , Xn are independent random variables, each with the density function f, show that the joint density of X(1), . . . , X(n) is

n!f(x1)f(x2) · · · f(xn), x1 < x2 < · · · < xn.

3. Solve Problem.3.8.74

Problem (3.8.74). Let U1, U2, and U3 be independent uniform random variables. a. Find the joint density of U(1), U(2), and U(3). b. The locations of three gas stations are independently and randomly placed along a mile of highway. What is the probability that no two gas stations are less than 1

3 mile apart?

4. Let X1, X2, . . . be a sequence of random variables. Prove that sup Xn and lim supn→∞ Xn are random variables.

5. Independent random variables X1, X2, . . . are identically distributed from N(0, σ 2).

(a) If σ2 = 1, then show that X21 ∼ χ2(1) ∼ gamma(1/2, 1/2). (b) Find the density of Z = (X1 + · · · + Xk)/

√ k(X21 + · · · + X2k).

(c) Conclude that the density of Z does not contain σ term.

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