Download Discrete Random Variables

Chapter 5
Discrete Random Variables
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved.
Discrete Random Variables
5.1 Two Types of Random Variables
5.2 Discrete Probability Distributions
5.3 The Binomial Distribution
5.4 The Poisson Distribution (Optional)
5-2
Two Types of Random Variables
• Random variable: a variable that assumes
numerical values that are determined by the
outcome of an experiment
– Discrete
– Continuous
• Discrete random variable: Possible values
can be counted or listed
– The number of defective units in a batch of 20
– A listener rating (on a scale of 1 to 5) in an
AccuRating music survey
5-3
Random Variables
Continued
• Continuous random variable: May
assume any numerical value in one or
more intervals
– The waiting time for a credit card
authorization
– The interest rate charged on a business
loan
5-4
Discrete Probability Distributions
• The probability distribution of a
discrete random variable is a table,
graph or formula that gives the
probability associated with each
possible value that the variable can
assume
• Notation: Denote the values of the random
variable by x and the value’s associated
probability by p(x)
5-5
Discrete Probability Distribution
Properties
1. For any value x of the random
variable, p(x)  0
2. The probabilities of all the events in
the sample space must sum to 1, that
is…
 px   1
all x
5-6
Expected Value of a Discrete Random
Variable
The mean or expected value of a
discrete random variable X is:
m X   x p x 
All x
m is the value expected to occur in the
long run and on average
5-7
Variance
• The variance is the average of the
squared deviations of the different
values of the random variable from the
expected value
• The variance of a discrete random
variable is:
2
X
   x  m X  p x 
2
All x
5-8
Standard Deviation
• The standard deviation is the square
root of the variance
X 
2
X
• The variance and standard deviation
measure the spread of the values of the
random variable from their expected
value
5-9
The Binomial Distribution
•
The binomial experiment…
1. Experiment consists of n identical trials
2. Each trial results in either “success” or “failure”
3. Probability of success, p, is constant from trial to
trial
–
The probability of failure, q, is 1 – p
4. Trials are independent
•
If x is the total number of successes in n
trials of a binomial experiment, then x is a
binomial random variable
5-10
Binomial Distribution
Continued
• For a binomial random variable x, the
probability of x successes in n trials is given
by the binomial distribution:
n!
px  =
p x q n- x
x!n - x !
– n! is read as “n factorial” and n! = n × (n-1) × (n-2)
× ... × 1
– 0! =1
– Not defined for negative numbers or fractions
5-11
Binomial Probability Table
Table 5.7(a) for n = 4, with x = 2 and p = 0.1
p = 0.1
values of p (.05 to .50)
x
0
1
2
3
4
0.05
0.8145
0.1715
0.0135
0.0005
0.0000
0.95
0.1
0.6561
0.2916
0.0486
0.0036
0.0001
0.9
0.15
0.5220
0.3685
0.0975
0.0115
0.0005
0.85
…
…
…
…
…
…
…
0.50
0.0625
0.2500
0.3750
0.2500
0.0625
0.50
4
3
2
1
0
x
values of p (.05 to .50)
P(x = 2) = 0.0486
5-12
Several Binomial Distributions
5-13
Mean and Variance of a Binomial
Random Variable
• If x is a binomial random variable with
parameters n and p (so q = 1 – p), then
– Mean m = n•p
– Variance 2x = n•p•q
– Standard deviation x = square root n•p•q
 X  npq
5-14