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Chapter 5
Discrete Random Variables
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Chapter Outline
5.1
5.2
5.3
5.4
5.5
5.6
Two Types of Random Variables
Discrete Probability Distributions
The Binomial Distribution
The Poisson Distribution (Optional)
The Hypergeometric Distribution
(Optional)
Joint Distributions and the Covariance
(Optional)
5-2
LO 5-1: Explain the
difference between a
discrete random
variable and a
continuous random
variable.
5.1 Two Types of Random
Variables
 Random variable: a variable that assumes numerical
values 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 rating on a scale of 1 to 5
 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
5-3
LO 5-2: Find a discrete
probability distribution
and compute its mean
and standard deviation.
5.2 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
 Denote the values of the random variable by
x and the value’s associated probability by
p(x)
5-4
LO 5-3: Use the
binomial distribution
to compute
probabilities.
5.3 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
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
5-5
LO 5-4: Use the
Poisson distribution to
compute probabilities
(Optional).
5.4 The Poisson Distribution
(Optional)
 Consider the number of times an event
occurs over an interval of time and assume
1.
2.
The probability of occurrence is the same for
any intervals of equal length
The occurrence in any interval is
independent of an occurrence in any nonoverlapping interval
 If x = the number of occurrences in an
interval, then x is a Poisson random variable
e  x
px  
x!
5-6
LO5-4
Poisson Probability Calculations
px  
p 0  
p 1 
p 2  
p 3 
p 4  
p 5 
e  x
x!
0
e .4 .4 
0!
1
e .4 .4 
1!
2
e .4 .4 
2!
3
e .4 .4 
3!
4
e .4 .4 
4!
5
e .4 .4 
5!
 .6703
 .2681
 .0536
 .0072
 .0007
 .0001
5-7
LO 5-5: Use the
hypergeometric
distribution to
compute probabilities
(Optional).
5.5 The Hypergometric
Distribution (Optional)
 Population consists of N items


r of these are successes
(N-r) are failures
 If we randomly select n items without
replacement, the probability that x of the n
items will be successes is given by the
hypergeometric probability formula
 r  N  r 
 

x nx 
P ( x)   
N
 
n
5-8
LO5-5
The Mean and Variance of a
Hypergeometric Random Variable
Mean
r 
 x  n 
N
Variance

2
x
r  N  n 
 r 
 n 1  

 N  N  N  1 
5-9
LO 5-6: Compute and
understand the
covariance between
two random variables
(Optional).
5.6 Joint Distributions and the
Covariance (Optional)
5-10
LO5-6
Covariance
 To measure the association between x and y,
can calculate the covariance between x and y
 Calculate (x-µx)(y-µy)=(x-.124)(y-.124) for each
combination of values of x and y

Note that .124 is the mean of both distributions
 Multiply each (x-µx)(y-µy) value by the
probability of p(x,y) and sum results
 The result is the covariance
 Denoted by σ2xy
5-11