5-1
l Chapter 5 l
Discrete Probability
Distributions
5.1 Binomial Probability Distribution
5.2 Poisson Probability Distribution
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
5-2
5.1 Binomial Probability Distribution
 An
experiment in which satisfied the following characteristic
is called a binomial experiment:




The random experiment consists of n identical trials.
Each trial can result in one of two outcomes, which we denote by
success, S or failure, F.
The trials are independent.
The probability of success is constant from trial to trial, we denote
the probability of success by p and the probability of failure is equal
to (1-p) = q.
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
5-3
5.1 Binomial Probability Distribution
 Examples:



No. of getting a head in tossing a coin 10 times.
No. of getting a six in tossing 7 dice.
A firm bidding for contracts will either get a contract or not
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
5-4
5.1 Binomial Probability Distribution
A
binomial experiment consist of n identical trial with
probability of success, p in each trial. The probability of x
success in n trials is given by
P( X  x)  nCx p x q n  x ; x  0,1, 2,..., n
 If
X~ B(n, p), then
  E ( X )  np
 2  npq   npq
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
5-5
5.1 Binomial Probability Distribution
 Given
that X~B (12, 0.4), find
a) P( X  2)
b) P( X  3)
c) P( X  4)
d) P(2  X  5)
e) E ( X )
f ) 2
Irwin/McGraw-Hill
Answer:
a) 0.0639
b) 0.1419
c) 0.2128
d) 0.4185
e) 4.8
f)
2.88
© Andrew F. Siegel, 1997 and 2000
5-6
5.1 Binomial Probability Distribution
 Cumulative
Binomial Distribution
i) P( X  x)  P( X  x)  P ( X  x  1)
ii) P( X  x)  P( X  x  1)
iii) P( X  x)  P( X  x  1)  1  P( X  x)
iv) P( X  x)  1  P ( X  x  1)
v) P ( x1  X  x2 )  P ( X  x2 )  P ( X  x1  1)
vi) P( x1  X  x2 )  P( X  x2  1)  P( X  x1 )
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
5-7
5.1 Binomial Probability Distribution
 Given
X~B (25, 0.15), using tables of Binomial probabilities,
find
a) P( X  5)
b) P( X  5)
c) P( X  7)
d) P( X  8)
e) P(5  X  8)
f ) P(2  X  10)
g) P(4  X  11)
Irwin/McGraw-Hill
Answer:
a) 0.8385
b) 0.6821
c) 0.0255
d) 0.0255
e) 0.3099
f) 0.7442
g) 0.3178
© Andrew F. Siegel, 1997 and 2000
5-8
5.1 Binomial Probability Distribution
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
5-9
5.2 Poisson Probability Distribution
A
random variable X has a Poisson distribution and it is
referred to as a Poisson random variable if and only if its
probability distribution is given by
e   x
P( X  x) 
for x  0,1, 2,3,..., n
x!

 (Lambda) is the long run mean number of events for the
specific time or space dimension of interest. A random
variable X having a Poisson distribution can also be written as
X ~ P0 ( )
with E ( X )   and   
2
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
5-10
5.2 Poisson Probability Distribution
 Poisson
distribution is the probability distribution of the
number of successes in a given space.
 Space can be dimensions, place or time or combination of
them.

a)
b)
c)
Examples:
No. of cars passing a toll booth in one hour.
No. defects in a square meter of fabric.
No. of network error experienced in a day.
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
5-11
5.2 Poisson Probability Distribution
 Given
that X ~ P0 (4.8) , find
a) P ( X  0)
b) P ( X  9)
c) P ( X  1)
Irwin/McGraw-Hill
Answer:
a) 0.0082
b) 0.0307
© Andrew F. Siegel, c)
19970.9918
and 2000
5-12
5.2 Poisson Probability Distribution
 Suppose
that the number of errors in a piece of software has a
Poisson distribution with parameter   3 . Find
a)
b)
c)
The probability that a piece of software has no errors.
The probability that there are three or more errors in piece of
software .
The mean and variance in the number of errors.
Answer:
a) 0.05
b) 0.577
c) ?
Irwin/McGraw-Hill
© Andrew F. Siegel, 1997 and 2000
5-13
5.2 Poisson Probability Distribution
 The
no. of points scored by Team A in 1 basketball match is
Poisson distributed with mean 2.4. What is the probability
that the team scores at least 16 points in
 4 matches
Irwin/McGraw-Hill
Answer:
0.0362
© Andrew F. Siegel, 1997
and 2000