Homework Assignments
Homework 3
 Suppose and are the indicator functions of events and , respectively. Find , and show that and are independent if and only if .
 Suppose is a ch.f. Show that is also a ch.f.
 Suppose and are jointly Gaussian. Use ch.f.s to show that .

Suppose
and
are jointly continuous. (a) Show that
 Suppose and are independent continuous r.v.s with c.d.f.s and , respectively. Suppose further that for all . Show that
 Prove Jensen's inequality for the case of simplefunction r.v.'s
 Prove the Schwartz inequality.
Problems from Grimmet & Stirzaker :
 Prob 2.7.4
 Prob 2.7.7. Hint: binomial distribution

Prob 2.7.9. Hint:
 Ex 3.3.1. Hint: Let , and .

Ex 3.4.1. Let
be the indicator function of the event that
the outcome of the
st toss is different from the outcome of
the
th toss. The number
of distinct runs is
. Observe that
and
are independent
if
. Show that

Ex. 3.4.2.
Hint: Let
, where
is the
number on the
th ball. Show that:
.
show that
. Hint:
 Ex 3.5.2. Hint: .
 Ex 3.6.5. Hint: , with equality if and only if .
Copyright 2008,
Todd Moon.
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.
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.
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