stat310

# Homework 06

This homework follows the standard late penalty: 0% if in the stat310 mailbox by Thursday 8 Mar 4pm, 10% by 5pm the following day, 100% otherwise. Please read the syllabus for other homework policies.

(Bonus homework worth half a normal homework)

1. (5 points) Let $$f(x, y) = c(x + 2xy + 2y)$$, $$x \in [0,1], y \in [0,1]$$

1. What is $$c$$ ?

2. What is $$F(x, y)$$?

2. (8 points). Which of the following bivariate pdfs represent the pdf of two independent pdfs? (You can assume $$x \in [0,1], y \in [0,1]$$ and you don’t need to find the values of any constants). Show your reasoning.

1. $$f(x, y) = c_1 e^{x+y}$$

2. $$f(x, y) = c_2 (x + y)$$

3. $$f(x, y) = c_3 (xy + x + y + 1)$$

4. $$f(x, y) = c_4 (x^2 y^2 + x^2 y + x^2)$$

3. (6 points) $$X \sim Unif(0, 10)$$, $$Y | X = x \sim Exp(\theta = x)$$. Find:

1. $$f(x, y)$$

2. $$f(y)$$

3. $$E(Y)$$