Assignment 4

Question 1 (Exercise 7.6)

Suppose that \(X\) and \(Y\) have joint probability density function \[f(x,y) = \begin{cases} 6e^{-(2x+3y)} &\text{if } x \ge 0\text{ and }y \ge 0 \\ 0 & \text{otherwise}. \end{cases}\] By integrating the joint probability density function, calculate:

1. \(\pr{X < 1/2, Y > 1/2}\), and

2. \(\pr{X > Y}\).

Question 2 (Exercise 8.7)

Let \(X \sim \text{U}(0, 2\pi)\). Find \(\expec{\sin X}\).

Question 3 (Exercise 8.8)

Let \(X \sim \text{U}(0, \pi)\). Find \(\expec{\sin X}\).

Question 4 (comparative judgement question)

Write a summary, of around forty words, explaining your current best understanding of what expectation is about. Feel free to use any examples or related functions to help you illustrate your explanation. The best summaries are ones that use key ideas, rather than repeating the original definition.

Next week, you will be asked to look at pairs of these summaries, submitted by your peers, and select the summary which you think best encompasses this idea.