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.
(4 pts) Find Var(aX+bY) and express it as simply as possible (don’t make any assumptions about X and Y)
Let \(X_1, X_2, …, X_n \) be iid \( \mbox{Poisson}(\lambda) \).
(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 pts) What is another mgf that \(\bar{X}_{n} \) should be close to? Why?
(2 pts) Compare the mgf of exact distribution (from a) to the approximate distribution (from b). What happens as \(n \rightarrow \infty \)?
Let \(X \) be a random variable with pdf \( f(x) = 630x^{4}(1-x)^{4}, x \in (0, 1) \)
(2 pts) Find E(X) and Var(X).
(2 pts) Obtain the lower bound given by Chebyshev’s inequality for \(P(0.2 < X < 0.8) \).
(1 pt) Compute the exact probability, \( P(0.2 < X < 0.8) \).
Extra credit: (5pts)
Attend Scott Berry’s lecture, March 19, 2012, 4-5pm in Duncan Hall’s McMurtry Auditorium.
Write 2 paragraphs about what you learned and how it relates to Stat310.