University of Illinois Spring 2005
ECE 413: Problem Set 2
Due: Wednesday February 2 at the beginning of class.
Reading: Ross, Chapter 1 and Chapter 2, Sections 1-5
Noncredit Exercises: DO NOT turn these in.
Chapter 1: Problems 1-5, 7, 9; Theoretical Exercises 4, 6, 13; Self-Test Problems 1-15.
Chapter 2: Problems 3, 4, 9, 10,11-14; Theoretical Exercises 1-3, 6, 7, 10, 11, 2,16, 19, 20;
Self-Test Problems 1-8
This Problem Set contains six problems
1. The binomial theorem avers that (1 + x)n=
n
X
k=0 n
kxkwhere n
k=n!
k!(n−k)! is
called a binomial coefficient.
(a) Show that n
k=n
n−kand that
n
X
k=0 n
k= 2n.
(b) n
kis also the number of events of size kdefined on a sample space of size n.
Use this notion to prove the results of part (a) without using the binomial theorem
or any explicit formula for n
k.
(c) Suppose that nis fixed. What choice(s) of kmaximize the value of n
k? For
which values of nis the maximum unique? If there is more than one maximum,
how many maxima can there be? Warning: You cannot use the usual calculus
trick of differentiating here because kis not a continuous variable.
(d) Write down the first five terms in the expansions of (1+x)nand (1−x)n(preferably
on two consecutive lines and with coefficients of xkon the two lines being one above
the other in vertical alignment.) Now show that exactly 2n−1of the 2nsubsets
of a sample space of size ncontain an even number of elements (the other 2n−1
subsets contain an odd number of elements). Hint: remember that 0 is an even
number and set x= 1 in (1 + x)n±(1 −x)n.
2. (a) What is the derivative of exp(−x2/2)?
(b) Find the value of Z∞
0
xexp(−x2/2) dx. Hint: read your answer of part (a).
(c) Find the value of Z∞
0Z∞
0
exp(−[x2+y2]/2) dx dy. Hint: convert to polar coor-
dinates and read your answer to part (b).
