Exercises: Conditional probability
54. Let R and G be the values when a red die and a green die are tossed, and let N be their
minimum. Find P(N = 2 | R = r) for r = 1,2,3,4,5,6.
Solution: Draw a conditional Venn diagram depicting the values of N. Here is

12 Important Continuous Random Variables
P(T > t) = e t
The exponential distribution as the continuous analog of the geometric distribution
If T has an exponential distribution then excess probabilities P(T > t) can be expressed as
P(T > t) = e t = (e)t =

Independence ( Quick start
Exercises: Independence Quick Start
73. In most probability models, independence is assumed, rather than proven, and the basis
for the assumption is the intuitive notion of independence discussed above. Consider the
following ex

Exercises: Events
16. Consider the random experiment in which a coin is tossed 5 times. Let S be the total
number of heads obtained, and let W be the number of tosses prior to the first head.
(a) Write down an event A which involves only the random variab

Exercises: Expectation, means, variances and covariances
36. Find the means and variances of the random variables having the following densities:
12 x 2 (1 x ) 0 x 1
(a) f(x) =
otherwise.
0
ln t 0 < t 1
(b) f(t) =
otherwise.
0
2 x 3
(c) f(x) =
0
x1
ot

Distribution functions
0
FY ( x) = 1 (1 x) 2
1
if x < 0
if 0 x 1
if x > 1.
and so has density
2(1 x) if 0 x 1
fY(x) = FY ( x) =
if x < 0 or x > 1.
0
The graphs of these densities are as follows:
2.
1.5
fX(x)
1.
.5
.
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0.6
0.8
1

Exercises: Bayes rule
136. In the example Testing for AIDS (p. 150), use a spreadsheet to graph the conditional
probability P(H = 1 | T = 1) as a function of the seroprevalence probability .
Solution: We wish to graph the function
0.934
= P(H=1|T=1)
0.93

11 Further Properties of Continuous Random Variables
The average distance to the city is approximately 2.77 miles. We shall derive the true
density of the distance Y to the city in a subsequent exercise. Both the relative frequency
per mile from the city

1 Basic Concepts
Track a newly sold car and observe the mileage and elapsed time until the first
breakdown.
We can define
T = the time until the first breakdown
Y = the distance driven before the first breakdown.
In practice, the time T would be measured

Density functions
Student: Well, Id have to get the digit 5, then the digit 0, then the digit 0, then .
well, an infinite string of 0s. The probability of that would be
111
10 10 10
which, I guess, is zero, isnt it?
Prof: Thats right. And the probabilit

Exercises: Covariance and correlation as measures of association
136. Show that Cov(X,X) = Var[X], and X,X = 1.
Solution:
X,X = Cov(X,X)
by definition
= E[(XX)(XX)] by definition
= E[(XX)2]
= Var[X]
by definition
2
= X
by definition
Then
X,X =
X,X
=
X X

Combinations
Exercises: Combinations
30. Consider the set cfw_a,b,c,d,e containing 5 elements. According to Theorem 7, there are
5
subsets of size k for k = 0,1,2,3,4,5. For each k, identify exactly what those subsets
k
are.
Solution:
5
k = 0: = 1 set, w

IEMS 202
Probability
Spring 2014
Kezban Yagci Sokat kezban.yagcisokat@u.northwestern.edu Tech C229
Edwin Shi zhenyushi2013@u.northwestern.edu
Instructor:
TA:
Lecture: MWF 9:00-9:50, Tech A110
Lab: Thursdays 2:00-2:50, 3:00-3:50, or 4:00-4:50 Tech C135
Off