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AMS 311
Lecture 12
March 2, 2000
Twenty Minute (Two Problem) Quiz next Thursday, March 9, 2000 from Chapter one
and two. One forty point proof question, and one thirty point problem.
Homework from Chapter Four due on March 9: P 141: 2, 8; p 148: 4, 7; p. 159: 2, 10,
12, 16*; p.168: 4, 7,; p.171: 2.
Chapter Four
Distribution Functions and Discrete Random Variables
4.1. Random Variables
Definition: Let S be the sample space of an experiment. A real-valued function
X : S  R is called a random variable of the experiment if, for each interval
I  R,{s: X ( s)  I } is an event.
Key points: objective numerical value obtained from running an experiment; intervals in
R have to be events (that is, must have a probability assigned by the probability measure).
4.2. Distribution Functions (I may use the term “cumulative distribution function”
abbreviated to cdf in class)
Definition: If X is a random variable, then the function F defined on (-, ) by
F (t )  P( X  t ) is called the distribution function of X.
1.
2.
3.
4.
F is nondecreasing.
limt  F (t )  1.
limt  F (t )  0.
F is right continuous.
Example Use of Distribution Functions in the Key Problem of Statistical Design
The faculty of a statistics department is considering using an eight question true-false
test to determine whether a student is a random guesser or is knowledgeable about
statistics. That is, they will present a student with eight true-false questions of equal
difficulty in random order. They will use the observed value of E8, the number of errors
that the student makes, as the basis for accepting or rejecting the null hypothesis H0 that
their student is a random guesser. They will reject H0 when they observe E8 1 and
accept H0 otherwise.
They computed F0, the cumulative distribution function of E8 under the null
hypothesis. They also computed F1, the cumulative distribution function of E8 for a
student who had a 0.90 chance of correctly answering each question. Table 1 contains
these values.
Table 1
Cumulative Distribution Function of E8
under H0 and for Knowledgeable Student
s
F0(s)
0
1
2
3
4
5
6
7
8
0.0039
0.0352
0.1445
0.3633
0.6367
0.8555
0.9648
0.9961
1.0000
F1(s)
0.4305
0.8131
0.9619
0.9950
0.9996
1.0000
1.0000
1.0000
1.0000
1. What is α, the probability of a Type I error, for this test of the null hypothesis?
2. What is β, the probability of a Type II error, for this eight question examination
administered to a student who has a 0.90 probability of correctly answering each
question?
4.3. Discrete Random Variables
Definition: The probability function (probability mass function pmf) of a random
variable X whose set of possible values is {x1 , x2 , x3 ,} is a function from R to R that
satisfies the following properties:
1. p( x)  0 if x  {x1 , x2 , x3 ,}.
2. p( xi )  P( X  xi )
3.


i 1
p( xi )  1.
Example problem: find the probability function (pmf) of F1.
4.4 Expectations of Discrete Random Variables
Definition: The expected value of a discrete random variable X with the set of possible
values A and the probability function p(x) is defined by
E ( X )   xp( x).
x A
We say that E(X) exists if this sum converges absolutely.