stat310

# Homework 08

This homework follows the standard late penalty: 0% if in the stat310 mailbox by Monday 26 Mar 4pm, 10% by 5pm the following day, 100% otherwise. Please read the syllabus for other homework policies.

1. (4 pts) Find Var(aX+bY) and express it as simply as possible (don’t make any assumptions about X and Y)

2. Let $$X_1, X_2, …, X_n$$ be iid $$\mbox{Poisson}(\lambda)$$.

1. (3 pts) What is the exact mgf of $$S_n = \sum_{i = 1}^n X_i$$. Does this represent a named distribution? What is the mgf of $$\bar{X}_{n}=\frac{S_{n}}{n}$$ ?

2. (2 pts) What is another mgf that $$\bar{X}_{n}$$ should be close to? Why?

3. (2 pts) Compare the mgf of exact distribution (from a) to the approximate distribution (from b). What happens as $$n \rightarrow \infty$$?

3. Let $$X$$ be a random variable with pdf $$f(x) = 630x^{4}(1-x)^{4}, x \in (0, 1)$$

1. (2 pts) Find E(X) and Var(X).

2. (2 pts) Obtain the lower bound given by Chebyshev’s inequality for $$P(0.2 < X < 0.8)$$.

3. (1 pt) Compute the exact probability, $$P(0.2 < X < 0.8)$$.

4. Extra credit: (5pts)

1. Attend Scott Berry’s lecture, March 19, 2012, 4-5pm in Duncan Hall’s McMurtry Auditorium.

2. Write 2 paragraphs about what you learned and how it relates to Stat310.