University of Illinois Spring 2008
ECE 313: Problem Set 3
Axioms of Probability
This Problem Set contains six problems
Due: Wednesday February 6 at the beginning of class.
Reading: Ross, Chapter 2, Sections 1-5, and Chapter 4
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, 10,11,12,14;
Theoretical Exercises 1-3, 6, 10, 11, 16, 19, 20; Self-Test Problems 1-8
1. For α > 0, the Gamma function Γ(α) is defined as Γ(α) = Z∞
0
tα−1exp(−t)dt.
(a) Show by integration that Γ(1) = 1.
(b) Use integration by parts to show that Γ(α+ 1) = αΓ(α), and use this to deduce
that Γ(n+ 1) = n!.
(c) If αis not an integer, show that Γ(α+ 1) = α(α−1)(α−2) ···Γ(α−bαc) where
bαcis the integer part of α. Note that 0 < α − bαc<1. For 0 < β < 1, Γ(β)
must be evaluated by numerical integration, except. . .
(d) show that Γ(1
2) = √π. Hint: make the change of variables √2t=x. Then show
that
Γ1
2=2Z∞
0Z∞
0
exp(−[x2+y2]/2) dx dy1/2
where the double integral can be evaluated by a change to polar coordinates as
in Problem 4(b) of Problem Set 1.
2. The binomial coefficient n
k=n!
k!(n−k)! denotes the number of subsets of size k
drawn from a set of size n.
(a) Give a combinatorial proof that m+n
k=
k
X
i=0 m
i n
k−i. Hint: How many
committees of size kcan be formed from a group of mmen and nwomen?
(b) What is the coefficient of xkin the polynomial (1 + x)m+n?
Now write (1 + x)m+n= (1 + x)m(1 + x)nand find the coefficient of xkon the
right hand side in terms of the coefficients of (1 + x)mand (1 + x)n. If you have
taken ECE 410, you might recognize a discrete convolution here . . . .
(c) Use the result of part (a) to show that 2n
n=
n
X
i=0 n
i2
.
3. Ross, Chapter 2, Problem 13 (page 57).
