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Homework 2

  1. Suppose $ X$ is a r.v. with c.d.f. $ F_X$ . Prove the following:
    1. $ F_X$ is nondecreasing.
    2. $ \lim_{a\rightarrow \infty} F_X(a) = 1$ .
    3. $ \lim_{a\rightarrow-\infty} F_X(a) = 0$ .
    4. $ F_X$ is right continuous.
    5. $ P(a < X \leq b) = F_X(b) - F_X(a)$< X \leq b) = F_X(b) - F_X(a)$" align="middle" border="0" height="37" width="265" /> if $ b > a$ a$" align="middle" border="0" height="34" width="48" /> .
    6. $ P(X=a) = F_X(a) - \lim_{b \rightarrow a^-} F_X(b).$
    Also, find expressions for $ P(a \leq X \leq b)$ , $ P(a \leq X < b)$< b)$" align="middle" border="0" height="37" width="119" /> and $ P(a < X < b)$< X < b)$" align="middle" border="0" height="37" width="119" /> in terms of $ F_X$ .
  2. Show that the following are valid p.m.f.s:
    1. Binomial: $ f_X(a) = n!/((n-a)!a!) \pi^a (1-\pi)^{n-a}$ if $ a\in\{0,1,\ldots,n\}$ .
    2. Poisson: $ f_X(a) = e^{-\lambda}\lambda^a/a!$ for $ a \in \{0,1,\ldots\}$ .
  3. Find the mean and variance of $ X$ when $ X$ is (a) $ \Nc(\mu,\sigma^2)$ ; (b) Binomial$ (n,\pi)$ ; (c) Poisson$ (\lambda)$ ; (d) Exponential$ (\lambda)$ . Do not use characteristic functions.
  4. Suppose that $ X$ and $ Y$ are jointly continuous. Show that

    $\displaystyle f_X(x) = \int_{-\infty}^\infty f_{XY}(x,y) dy \qquad x \in \Rbb. $

  5. Suppose that $ X$ and $ Y$ are jointly Gaussian with parameters $ \mu_x,\sigma_x^2, \mu_y, \sigma_y^2, \rho$ . Show that $ X \sim \Nc(\mu_x, \sigma_x^2)$ .
  6. Suppose $ X \sim \Nc(0,1)$ , and define $ Y=X^2$ . Are $ X$ and $ Y$ uncorrelelated? Are $ X$ and $ Y$ independent? Find the p.d.f. of $ Y$ . Are $ X$ and $ Y$ jointly continuous?
  7. Show that $ \cov(aX+b,cY+d) = ac \cov(X,Y)$ .
  8. Suppose $ X \sim \Nc(0,\sigma^2)$ . Use the ch.f. of $ X$ to find an expression for $ E[X^n]$ , $ n \in \Zbb^+$ .

Problems from the Grimmett & Stirzaker text:

  1. Ex 1.4.4
  2. Ex 1.4.5. Hint: Let $ C_i$ be the event that the $ i$ th door conceals the car, let $ G$ be the event that you see a goat, and let $ B$ be the event that you see Bill.
  3. Ex 1.5.1.
  4. Ex 1.5.2
  5. Ex 1.5.7.
  6. Prob. 1.8.5
  7. Prob. 1.8.6.
  8. Prob. 1.8.19.
  9. Prob. 1.8.20.
  10. Prob. 1.8.30.
Copyright 2008, Todd Moon. Cite/attribute Resource. admin. (2006, June 13). Homework Assignments. Retrieved November 24, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/hw2.html. This work is licensed under a Creative Commons License. Creative Commons License
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