stat310

# Homework 04

This homework follows the standard late penalty: 0% if in the stat310 mailbox by Thursday 9 Feb 4pm, 10% by 5pm the following day, 100% otherwise. Please read the syllabus for other homework policies.

1. (3 pts) For each of the following mgfs, identify the distribution of the corresponding random variable (including it’s parameters)

1. $$M_{X}(t) = 2^{e^{t}-1}$$
2. $$M_{Y}(t) = 0.5(1+e^{t})$$
3. $$M_{Z}(t) = \left (\frac{1-\lambda e^{t}}{1-\lambda}\right )^{-\alpha}$$
2. (8 pts) Given the pmf $$f(k) = \frac{-1}{ln(1-p)}\frac{p^{k}}{k}, k = 1, 2, …. \infty, 0 < p < 1$$

1. Find the mgf
2. Use the mgf to find the mean & variance.
3. (8 pts) Given $$f(x) = \frac{c}{(1+x^{2})^{2}}$$ $$-1 < x < 1$$

1. Find the value of $$c$$ to make it a pdf.
2. Argue that the mean should be 0 without performing any calculation.
3. Find the mean and variance without using the mgf.
4. (4 pts) Given $$F(x) = (1 - cos(x))/2 , x \in [0, \pi]$$; Find:

1. $$P(x < \pi/2)$$
2. $$P(\pi/4 < x < 3\pi/4)$$
3. the pdf.