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Homework 3

  1. Suppose $ X$ and $ Y$ are the indicator functions of events $ A$ and $ B$ , respectively. Find $ \rho(X,Y)$ , and show that $ X$ and $ Y$ are independent if and only if $ \rho(X,Y) = 0$ .
  2. Suppose $ \phi(u)$ is a ch.f. Show that $ \vert\phi(u)\vert^2$ is also a ch.f.
  3. Suppose $ X$ and $ Y$ are jointly Gaussian. Use ch.f.s to show that $ \rho(X,Y) = \rho$ .
  4. Suppose $ X$ and $ Y$ are jointly continuous. (a) Show that

    $\displaystyle F_{Y\vert X}(b\vert x) = \int_{-\infty}^b \frac{f_{XY}(x,y)}{f_X(x)}\,dy
$

    and thus that

    $\displaystyle f_{Y\vert X}(y\vert x) = \frac{f_{XY}(x,y)}{f_X(x)}
$

    (b) Suppose $ \int_{-\infty}^\infty \vert y\vert f_{Y\vert X}(y\vert x) dy < \infty$ . Show that $ E[Y\vert X=x] = \int_{-\infty}^\infty y f_{Y\vert X}(y\vert x) dy.$
  5. Suppose $ X$ and $ Y$ are independent continuous r.v.s with c.d.f.s $ F_X$ and $ F_Y$ , respectively. Suppose further that $ F_X(b)
\geq F_Y(b)$ for all $ b \in \Rbb$ . Show that $ P(X \geq Y) \leq
1/2.$
  6. Prove Jensen's inequality for the case of simple-function r.v.'s
  7. Prove the Schwartz inequality.

Problems from Grimmet & Stirzaker :

  1. Prob 2.7.4
  2. Prob 2.7.7. Hint: binomial distribution
  3. Prob 2.7.9. Hint:

    \begin{displaymath}P(X^- \leq x) =
\begin{cases}
0 & x < 0, \\
1- \lim_{y \uparrow -x} F(y) & x \geq 0.
\end{cases}\end{displaymath}

  4. Ex 3.3.1. Hint: Let $ p_X(-1) = \frac{1}{9}$ , $ p_X(\frac{1}{2})
= \frac{4}{9}$ and $ p_X(2) = \frac{4}{9}$ .
  5. Ex 3.4.1. Let $ I_j$ be the indicator function of the event that the outcome of the $ (j+1)$ st toss is different from the outcome of the $ j$ th toss. The number $ R$ of distinct runs is $ R = 1 +
\sum_{j=1}^{n-1} I_j$ . Observe that $ I_j$ and $ I_k$ are independent if $ \vert j-k\vert>1$ . Show that

    $\displaystyle E[(R-1)^2] = (n-1)E[I_1] + 2(n-2)E[I_1I_2] + ( (n-1)^2 - (n-1) -
2(n-2)) E[I_1]^2.
$

    Show that $ E[I_1I_2] = p^2 q + pq^2 = pq.$
  6. Ex. 3.4.2. Hint: Let $ T = \sum_{i=1}^k X_i$ , where $ X_i$ is the number on the $ i$ th ball. Show that: $ E[T] = \frac{1}{2} k(n+1)$ . show that $ E[T^2] = \frac{1}{6}k(n+1)(2n+1) +
\frac{1}{12}k(k-1)(3n+2)(n+1)$ . Hint:

    $\displaystyle \sum_{k=1}^N k = \frac{n(n+1)}{2} \qquad \qquad
\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}.
$

    $\displaystyle \sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2.
$

  7. Ex 3.5.2. Hint: $ P(H=x) = \sum_{n=x}^\infty P(H=x\vert N=n)P(N=n)$ .
  8. Ex 3.6.5. Hint: $ \log y \leq y-1$ , with equality if and only if $ y=1$ .

Copyright 2008, Todd Moon. Cite/attribute Resource . admin. (2006, June 13). Homework Assignments. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/hw3.html. This work is licensed under a Creative Commons License Creative Commons License