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.