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Chapter 5
Discrete Random Variables
and Probability
Distributions
©
Random Variables
A random variable is a variable that
takes on numerical values determined
by the outcome of a random experiment.
Discrete Random Variables
A random variable is discrete if it
can take on no more than a
countable number of values.
Discrete Random Variables
(Examples)
1.
2.
3.
4.
The number of defective items in a sample of
twenty items taken from a large shipment.
The number of customers arriving at a check-out
counter in an hour.
The number of errors detected in a corporation’s
accounts.
The number of claims on a medical insurance
policy in a particular year.
Continuous Random Variables
A random variable is continuous if
it can take any value in an interval.
Continuous Random Variables
(Examples)
1.
2.
3.
4.
5.
The income in a year for a family.
The amount of oil imported into the U.S. in a
particular month.
The change in the price of a share of IBM common
stock in a month.
The time that elapses between the installation of a
new computer and its failure.
The percentage of impurity in a batch of
chemicals.
Discrete Probability
Distributions
The probability distribution function (DPF),
P(x), of a discrete random variable expresses
the probability that X takes the value x, as a
function of x. That is
P( x)  P( X  x), for all values of x.
Discrete Probability
Distributions
(Example 5.1)
Graph the probability distribution function for
the roll of a single six-sided die.
P(x)
1/6
1
2
3
4
Figure 5.1
5
6
x
Required Properties of Probability
Distribution Functions of Discrete
Random Variables
i.
ii.
Let X be a discrete random variable with
probability distribution function, P(x). Then
P(x)  0 for any value of x
The individual probabilities sum to 1; that is
 P( x)  1
x
Where the notation indicates summation over
all possible values x.
Cumulative Probability Function
The cumulative probability function, F(x0),
of a random variable X expresses the
probability that X does not exceed the value
x0, as a function of x0. That is
F ( x0 )  P( X  x0 )
Where the function is evaluated at all values x0
Derived Relationship Between Probability
Function and Cumulative Probability
Function
Let X be a random variable with probability function
P(x) and cumulative probability function F(x0). Then
it can be shown that
F ( x0 )   P( X )
x  x0
Where the notation implies that summation is over all
possible values x that are less than or equal to x0.
Derived Properties of Cumulative
Probability Functions for Discrete
Random Variables
i.
ii.
Let X be a discrete random variable with
a cumulative probability function, F(x0).
Then we can show that
0  F(x0)  1 for every number x0
If x0 and x1 are two numbers with x0 < x1,
then F(x0)  F(x1)
Expected Value
The expected value, E(X), of a discrete random
variable X is defined
E ( X )   xP( x)
x
Where the notation indicates that summation
extends over all possible values x.
The expected value of a random variable is
called its mean and is denoted x.
Expected Value: Functions of
Random Variables
Let X be a discrete random variable with
probability function P(x) and let g(X) be
some function of X. Then the expected
value, E[g(X)], of that function is defined as
E[ g ( X )]   g ( x) P( x)
x
Variance and Standard Deviation
Let X be a discrete random variable. The
expectation of the squared discrepancies about
the mean, (X - )2, is called the variance,
denoted 2x and is given by
 x2  E ( X   x ) 2   ( x   x ) 2 P( x)
x
The standard deviation, x , is the positive
square root of the variance.
Variance
(Alternative Formula)
The variance of a discrete random variable X
can be expressed as
  E( X )  x
2
x
2
2
  x P( x)   x
2
x
2
Expected Value and Variance for
Discrete Random Variable Using
Microsoft Excel
(Figure 5.4)
Sales
P(x)
0
1
2
3
4
5
Mean
0.15
0.3
0.2
0.2
0.1
0.05
0
0.3
0.4
0.6
0.4
0.25
1.95
Expected Value = 1.95
Variance
0.570375
0.27075
0.0005
0.2205
0.42025
0.465125
1.9475
Variance = 1.9475
Summary of Properties for Linear
Function of a Random Variable
Let X be a random variable with mean x , and
variance 2x ; and let a and b be any constant fixed
numbers. Define the random variable Y = a + bX.
Then, the mean and variance of Y are
and
Y  E(a  bX )  a  b X

2
Y
 Var(a  bX )  b 
2
so that the standard deviation of Y is
Y  b X
2
X
Summary Results for the Mean and
Variance of Special Linear Functions
a)
Let b = 0 in the linear function, W = a + bX. Then W
= a (for any constant a).
E (a)  a
and
Var (a)  0
If a random variable always takes the value a, it will have
a mean a and a variance 0.
b) Let a = 0 in the linear function, W = a + bX. Then W
= bX.
E (bX )  b X
and
Var(a)  b 2 X2
Mean and Variance of Z
Let a = -X/X and b = 1/ X in the linear function
Z = a + bX. Then,
Z  a  bX 
so that
 X  X
E
 X
and
X  X
X

X
1
  

X  0
X X

 X  X
Var
 X
 1 2
  2  X  1
 X
Bernoulli Distribution
A Bernoulli distribution arises from a random
experiment which can give rise to just two possible
outcomes. These outcomes are usually labeled as
either “success” or “failure.” If  denotes the
probability of a success and the probability of a
failure is (1 -  ), the the Bernoulli probability
function is
P(0)  (1   ) and P(1)  
Mean and Variance of a
Bernoulli Random Variable
The mean is:
 X  E ( X )   xP( x)  (0)(1   )  (1)  
X
And the variance is:
 X2  E[( X   X ) 2 ]   ( x   X ) 2 P( x)
X
 (0   ) (1   )  (1   )    (1   )
2
2
Sequences of x Successes in n
Trials
The number of sequences with x successes in n
independent trials is:
n!
C 
x!(n  x)!
n
x
Where n! = n x (x – 1) x (n – 2) x . . . x 1 and 0! = 1.
These C xn sequencesare mutually exclusive,
since no two of them can occur at the same time.
Binomial Distribution
Suppose that a random experiment can result in two possible
mutually exclusive and collectively exhaustive outcomes, “success”
and “failure,” and that  is the probability of a success resulting in a
single trial. If n independent trials are carried out, the distribution
of the resulting number of successes “x” is called the binomial
distribution. Its probability distribution function for the binomial
random variable X = x is:
P(x successes in n independent trials)=
n!
x
( n x )
P( x) 
 (1   )
x!(n  x)!
for x = 0, 1, 2 . . . , n
Mean and Variance of a Binomial
Probability Distribution
Let X be the number of successes in n independent
trials, each with probability of success . The x follows
a binomial distribution with mean,
 X  E( X )  n
and variance,
  E[( X   ) ]  n (1   )
2
X
2
Binomial Probabilities
- An Example –
(Example 5.7)
An insurance broker, Shirley Ferguson, has five contracts,
and she believes that for each contract, the probability of
making a sale is 0.40.
What is the probability that she makes at most one sale?
P(at most one sale) = P(X  1) = P(X = 0) + P(X = 1)
= 0.078 + 0.259 = 0.337
5!
P(no sales)  P(0) 
(0.4) 0 (0.6) 5  0.078
0!5!
5!
P(1 sale)  P(1) 
(0.4)1 (0.6) 4  0.259
1!4!
Binomial Probabilities, n = 100,  =0.40
(Figure 5.10)
Sample size
100
Probability of success
0.4
Mean
40
Variance
24
Standard deviation
4.898979
Binomial Probabilities Table
X
36
37
38
39
40
41
42
43
P(X)
0.059141
0.068199
0.075378
0.079888
0.081219
0.079238
0.074207
0.066729
P(<=X)
0.238611
0.30681
0.382188
0.462075
0.543294
0.622533
0.69674
0.763469
P(<X)
0.179469
0.238611
0.30681
0.382188
0.462075
0.543294
0.622533
0.69674
P(>X)
0.761389
0.69319
0.617812
0.537925
0.456706
0.377467
0.30326
0.236531
P(>=X)
0.820531
0.761389
0.69319
0.617812
0.537925
0.456706
0.377467
0.30326
Hypergeometric Distribution
Suppose that a random sample of n objects is chosen from a
group of N objects, S of which are successes. The distribution
of the number of X successes in the sample is called the
hypergeometric distribution. Its probability function is:
C xS CnNxS
P( x) 
CnN
S!
( N  S )!

x!( S  x)! (n  x)!( N  S  n  x)!

N!
n!( N  n)!
Where x can take integer values ranging from the larger of 0
and [n-(N-S)] to the smaller of n and S.
Poisson Probability Distribution
Assume that an interval is divided into a very large
number of subintervals so that the probability of the
occurrence of an event in any subinterval is very
small. The assumptions of a Poisson probability
distribution are:
1) The probability of an occurrence of an event is
constant for all subintervals.
2) There can be no more than one occurrence in each
subinterval.
3) Occurrences are independent; that is, the number of
occurrences in any non-overlapping intervals in
independent of one another.
Poisson Probability Distribution
The random variable X is said to follow the Poisson
probability distribution if it has the probability function:
where
e   x
P( x) 
, for x  0, 1,2,...
x!
P(x) = the probability of x successes over a given period of
time or space, given 
 = the expected number of successes per time or space
unit;  > 0
e
= 2.71828 (the base for natural logarithms)
The mean and variance of the Poisson probability distribution are:
 x  E ( X )   and  x2  E[( X   ) 2 ]  
Partial Poisson Probabilities for  = 0.03
Obtained Using Microsoft Excel PHStat
(Figure 5.14)
Poisson Probabilities Table
X
P(X)
0
0.970446
1
0.029113
2
0.000437
3
0.000004
4
0.000000
P(<=X)
0.970446
0.999559
0.999996
1.000000
1.000000
P(<X)
0.000000
0.970446
0.999559
0.999996
1.000000
P(>X)
0.029554
0.000441
0.000004
0.000000
0.000000
P(>=X)
1.000000
0.029554
0.000441
0.000004
0.000000
Poisson Approximation to the
Binomial Distribution
Let X be the number of successes resulting from n independent
trials, each with a probability of success, . The distribution of
the number of successes X is binomial, with mean n. If the
number of trials n is large and n is of only moderate size
(preferably n  7), this distribution can be approximated by the
Poisson distribution with  = n. The probability function of the
approximating distribution is then:
e  n (n ) x
P( x) 
, for x  0, 1,2,...
x!
Joint Probability Functions
Let X and Y be a pair of discrete random variables.
Their joint probability function expresses the
probability that X takes the specific value x and
simultaneously Y takes the value y, as a function of
x and y. The notation used is P(x, y) so,
P ( x, y )  P ( X  x  Y  y )
Joint Probability Functions
Let X and Y be a pair of jointly distributed random
variables. In this context the probability function of the
random variable X is called its marginal probability
function and is obtained by summing the joint
probabilities over all possible values; that is,
P ( x )   P ( x, y )
y
Similarly, the marginal probability function of the
random variable Y is
P ( y )   P ( x, y )
x
Properties of Joint Probability
Functions
 Let X and Y be discrete random variables
with joint probability function P(x,y). Then
1. P(x,y)  0 for any pair of values x and y
2. The sum of the joint probabilities P(x, y)
over all possible values must be 1.
Conditional Probability
Functions
Let X and Y be a pair of jointly distributed discrete random
variables. The conditional probability function of the random
variable Y, given that the random variable X takes the value x,
expresses the probability that Y takes the value y, as a function of
y, when the value x is specified for X. This is denoted P(y|x), and
so by the definition of conditional probability:
P( x, y )
P( y | x) 
P( x)
Similarly, the conditional probability function of X, given Y = y
is:
P( x, y )
P( x | y ) 
P( y )
Independence of Jointly
Distributed Random Variables
The jointly distributed random variables X and Y are
said to be independent if and only if their joint
probability function is the product of their marginal
probability functions, that is, if and only if
P( x, y )  P( x) P( y ) for all possible pairs of values x and y.
And k random variables are independent if and only if
P( x1 , x2 ,, xk )  P( x1 ) P( x2 ) P( xk )
Expected Value Function of Jointly
Distributed Random Variables
Let X and Y be a pair of discrete random variables with
joint probability function P(x, y). The expectation of
any function g(x, y) of these random variables is
defined as:
E[ g ( X , Y )]   g ( x, y ) P( x, y )
x
y
Stock Returns, Marginal Probability,
Mean, Variance
(Example 5.16)
Y Return
X
Return
0%
0%
5%
10%
15%
0.0625
0.0625
0.0625
0.0625
5%
0.0625
0.0625
0.0625
0.0625
10%
0.0625
0.0625
0.0625
0.0625
15%
0.0625
0.0625
0.0625
0.0625
Table 5.6
Covariance
Let X be a random variable with mean X , and let Y be a
random variable with mean, Y . The expected value of (X
- X )(Y - Y ) is called the covariance between X and Y,
denoted Cov(X, Y).
For discrete random variables
Cov( X , Y )  E[( X   X )(Y  Y )]   ( x   x )( y   y ) P( x, y )
x
y
An equivalent expression is
Cov( X , Y )  E ( XY )   x  y   xyP( x, y )   x  y
x
y
Correlation
Let X and Y be jointly distributed random variables.
The correlation between X and Y is:
  Corr ( X , Y ) 
Cov( X , Y )
 X Y
Covariance and Statistical
Independence
If two random variables are statistically
independent, the covariance between them is 0.
However, the converse is not necessarily true.
Portfolio Analysis
The random variable X is the price for stock A and the
random variable Y is the price for stock B. The market
value, W, for the portfolio is given by the linear function,
W  aX  bY
Where, a, is the number of shares of stock A and, b, is the
number of shares of stock B.
Portfolio Analysis
The mean value for W is,
W  E[W ]  E[aX  bY ]
 a X  bY
The variance for W is,
  a   b   2abCov( X , Y )
2
W
2
2
X
2
2
Y
or using the correlation,
  a   b   2abCorr ( X , Y ) X  Y
2
W
2
2
X
2
2
Y
Key Words
 Bernoulli Random
Variable, Mean and
Variance
 Binomial Distribution
 Conditional Probability
Function
 Continuous Random
Variable
 Correlation
 Covariance
 Cumulative Probability
Function
 Differences of Random
Variables
 Discrete Random Variable
 Expected Value
 Expected Value: Functions
of Random Variables
 Expected Value: Function
of Jointly Distributed
Random Variable
 Hypergeometric
Distribution
 Independence of Jointly
Distributed Random
Variables
Key Words
(continued)
 Joint Probability Function
 Marginal Probability
Function
 Mean of Binomial
Distribution
 Mean: Functions of
Random Variables
 Poisson Approximation
to the Binomial
Distribution
 Poisson Distribution
 Portfolio Analysis
 Portfolio, Market Value
 Probability Distribution
Function
 Properties: Cumulative
Probability Functions
 Properties: Joint
Probability Functions
 Properties: Probability
Distribution Functions
 Random Variable
Key Words
(continued)
 Relationships: Probability
Function and Cumulative
Probability Function
 Standard Deviation:
Discrete Random
Variable
 Sums of Random
Variables
 Variance: Binomial
Distribution
 Variance: Discrete
Random Variable
 Variance: Discrete
Random Variable
(Alternative Formula)
 Variance: Functions of
Random Variables