Transcript Discrete Random Variables

Chapter 5
Discrete Random Variables
McGraw-Hill/Irwin
Copyright © 2014 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)
5.6 Joint Distributions and the Covariance
(Optional)
5-2
LO5-1: 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 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

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
LO5-2: 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
LO5-2
Discrete Probability Distribution Properties
1.
2.
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
LO5-2
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
LO5-2
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
LO5-2
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
5-8
LO5-3: Use the binomial
distribution to compute
probabilities.
5.3 The Binomial Distribution

The binomial experiment
characteristics…
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
5-9
LO5-3
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
5-10
LO5-3
Binomial Probability Table
p = 0.1
P(x = 2) = 0.0486
Table 5.4 (a) for n = 4, with x = 2 and p = 0.1
5-11
LO5-3
Several Binomial Distributions
Figure 5.7
5-12
LO5-3
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
5-13
LO5-4: Use the Poisson
distribution to compute
probabilities (Optional).
5.4 The Poisson Distribution

Consider the number of times an event
occurs over an interval of time or space,
and assume that
1. The probability of occurrence is the same for
any intervals of equal length
2. 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
5-14
LO5-4
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)
5-15
LO5-4
Poisson Probability Table
μ = 0.4
P( x  3) 
Table 5.6
e 0.4 (0.4)3
 0.0072
3!
5-16
LO5-4
Poisson Probability Calculations
Table 5.7
5-17
LO5-4
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
5-18
LO5-4
Several Poisson Distributions
Figure 5.10
5-19
LO5-5: 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
5-20
LO5-5
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 
5-21
LO5-5
Hypergeometric Example




Population of six stocks
Four have positive returns
We randomly select three stocks
Find P(x=2), mean, and variance
 r  N  r   4  2 
 
   
x
n

x
2  1  62





P( x  2) 


 0.6
20
N
 6
 
 
n
 3
 r  4
  3   2
N
  6
r  N  n   4  4  6  3 
 r 
 n 1  
  3 1  
  0.4
 N  N  N  1   6  6  6  1 
m x  n
 2x
5-22
LO5-6: Compute and
understand the
covariance between two
random variables
(Optional).
5.6 Joint Distributions and the
Covariance (Optional)
5-23
LO5-6
Calculating Covariance

To calculate covariance, calculate:
(x–μx)(y–μy)p(x,y)
for each combination of x and y
 Example on prior slide yields –0.0318
 A negative covariance says that as x
increases, y tends to decrease in a linear
fashion
 A positive covariance says that as x
increases, y tends to increase in a linear
fashion
5-24
LO5-6
Four Properties of Expected Values and
Variances
If a is a constant and x is a random
variable, then μax=aμx
2. If x1,x2,…,xn are random variables, then
μ(x1,x2,…,xn)= μx1 + μx2 + … + μxn
3. If a is a constant and x is a random
variable, then σ2ax=a2σ2x
4. If x1,x2,…,xn are statistically
independent random variables, then
the covariance is zero
1.
◦
Also, σ2(x1,x2,…,xn)= σ2x1+ σ2x2+…+ σ2xn
5-25