Transcript Discrete Random Variables - McGraw Hill Higher Education

Chapter 5
Discrete Random Variables
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
Discrete Random Variables
5.1
5.2
5.3
5.4
5.5
Two Types of Random Variables
Discrete Probability Distributions
The Binomial Distribution
The Poisson Distribution (Optional)
The Hypergeometric Distribution
(Optional)
5-2
LO 1: Explain the
difference between a
discrete random
variable and a
continuous random
variable.

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


5.1 Two Types of
Random Variables
The number of defective units in a batch of 20
Continuous random variable: May assume any
numerical value in one or more intervals

The waiting time for a credit card authorization
5-3
LO 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
•
Notation: Denote the values of the random
variable by x and the value’s associated
probability by p(x)
5-4
LO2
1.
2.
Discrete Probability Distribution
Properties
For any value x of the random variable, p(x)
0
The probabilities of all the events in the
sample space must sum to 1, that is…
 px   1
all x
5-5
LO2
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
5-6
LO2
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
5-7
LO 3: Use the binomial
distribution to compute
probabilities.
The binomial experiment characteristics…

1.
2.
3.
–
4.

5.3 The Binomial
Distribution
Experiment consists of n identical trials
Each trial results in either “success” or “failure”
Probability of success, p, is constant from trial to trial
The probability of failure, q, is 1 – p
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
5-8
LO3
Binomial Distribution

Continued
For a binomial random variable x, the probability of x
successes in n trials is given by the binomial
distribution:
n!
x n- x
px  =
p q
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
5-9
LO3

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
5-10
LO 4: Use the Poisson
distribution to compute
probabilities (optional).
Consider the number of times an event occurs
over an interval of time or space, and assume that

1.
2.

5.4 The Poisson
Distribution
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 non-overlapping interval
If x = the number of occurrences in a specified
interval, then x is a Poisson random variable
5-11
LO4
The Poisson Distribution
Continued


Suppose μ is the mean or expected number
of occurrences during a specified interval
The probability of x occurrences in the
interval when μ are expected is described by
the Poisson distribution
px  


e
m
m
x!
x
where x can take any of the values x = 0,1,2,3, …
and e = 2.71828 (e is the base of the natural logs)
5-12
LO4

Mean and Variance of a Poisson
Random Variable
If x is a Poisson random variable with
parameter m, then



Mean mx = m
Variance 2x = m
Standard deviation x is square root of variance
2x
5-13
LO 5: Use the
hypergeometric
distribution to compute
probabilities (optional).

Population consists of N items



5.5 The Hypergometric
Distribution (Optional)
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
5-14
LO5
The Mean and Variance of a
Hypergeometric Random Variable
Mean
r 
m x  n 
N
Variance

2
x
r  N  n 
 r 
 n 1  

 N  N  N  1 
5-15