Transcript Chapter 6. Discrete Probability Distributions

Chapter 6. Discrete
Probability Distributions
What Is Probability Distribution?
Ir. Muhril A., M.Sc., Ph.D.
1
What Is A Probability Distributions

Probability distributions gives the entire range
of values that can occur based on an
experiment.

Probability distributions:
A listing of all the outcomes of an experiment
and the probability associated with each
outcome.
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
2
Example:
Suppose we are interested in the number of
heads showing face up on three tosses of a coin.
This is the experiment. The possible results are:
zero heads,
One head,
Two heads, and
Three heads.
What is the probability distribution for the number of
heads?
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
3
The Results Are:
Coin Toss
Possible Result
First
Second
Third
Number Of
Heads
1
T
T
T
0
2
T
T
H
1
3
T
H
T
1
4
T
H
H
2
5
H
T
T
1
6
H
T
H
2
7
H
H
T
2
8
H
H
H
3
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
4
Probability Distribution For The Event Of Zero, One, Two, And
Three Heads From Three Tosses Of Coin
Number Of Heads (X)
0
Probability Of Outcome ,
P(X)
1/8 = 0.125
1
3/8 = 0.375
2
3/8 = 0.375
3
1/8 = 0.125
Total
8/8 = 1.000
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
5
What Is A Probability Distributions
(continued)

Character Of A Probability Distribution:
1. The probability of a particular outcome
is between 0 and 1 inclusive.
2. The outcome are mutually exclusive.
3. The sum of the probabilities of the
various events is equal to 1.
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
6
Random Variables

Random Variable:
a quantity resulting from an experiment that,
by chance, can assume different values.

A random variable may be either discrete or
continuous.
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
7
Random Variables (continued)

Discrete Random Variable:
- that can assume only certain clearly
separated values.

Continuous Random Variable:
- can assume one of an infinitely large
number of values, within certain
limitations.
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
8
The Mean, Variance, And Standard Deviation Of
A Discrete Probability Distributions

Mean:
- represent the central location of a
probability distributions.
- the long run average value of the random
variable.
- referred to as its expected value.
Equation 6-1 Page 185 (Lind).
m = S{X.P(X)}
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
9
The Mean, Variance, And Standard Deviation Of
A Discrete Probability Distributions (continued)

Variance And Standard Deviation:
Equation 6-2 Page 185 (Lind).
s2 = S{(X – m)2P(X)}
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
10
Example:
John Ragsdale sells new cars for Pelican Ford.
John usually sells the largest number of cars on
Saturday. He has developed the following
probability distribution for the number of cars he
expects to sell on a particular Saturday
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
11
Example:
Number Of Cars Sold (X)
Probability, P(X)
0
0.10
1
0.20
2
0.30
3
0.30
4
0.10
Total
1.00
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
12
Example:

On a typical Saturday, how many cars does
John expect to sell?
m = S{X.P(X)}
= 0(0.10) + 1(0.20) + 2(0.30) + 3(0.30) +
4(0.10)
= 2.1
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
13
Binomial Probability Distribution



Widely occurring discrete probability distribution.
There are only two possible outcomes on a particular trial of an
experiment.
The outcomes are mutually exclusive.
Equation 6-3 Page 190 (Lind).
P(X) = nCx px (1-p)n-x
P(X) = binomial probability
n = the number of trials
C = combination
x = the random variable defined as the number of successes
p = the probability of a success on each trial
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
14
Binomial Probability Distribution
(continued)

Mean Of A Binomial Distribution:
- equation 6-4 Page 191 (Lind).
m = np

Variance Of A Binomial Distribution:
- equation 6-5 Page 191 (Lind).
s2 = np(1-p)
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
15
Example:
There are five flights daily from Pittsburg via US
Airways into the Bradford, Pennsylvania,
Regional Airport. Suppose the probability that
any flight arrives late is 0.20.
- What is the probability that none of the flight
are late today?
- What is the probability that exactly one of the
flights is late today?
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
16
Example:
p = 0.20
n=5
x = the number of successes
case a success is a plane that arrives
late.
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
17
Example

No late arrivals (x = 0)
P(0) = 5C0(0.20)0(1-0.20)5-0 = 0.3277

The probability that exactly one of the five
flights will arrive late today is:
P(1) = 5C1(0.20)1(1-0.20)5-1 = 0.4096
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
18
Hypergeometric Probability
Distribution

An outcome on each trial of an experiment is classified into
one of two mutually exclusive categories, a success or a
failure.

The random variable is the number of successes in a fixed
number of trials.

The trials are not independent.

We assume that we sample from a finite population without
replacement and n/N>0.05.
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
19

Hypergeometric Distribution:
P(x) = [(sCx) (N-sCn-x)] / (NCn)
s = the number of successes in the population
C = combination
x = the number of successes in the sample
N = the size of population
n = the size of the sample or the number of
trials
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
20
Poisson Probability Distribution

Describes the number of times some event occurs
during a specified interval.

The interval may be time, distance, area, or volume.
Equation 6-7 Page 203 (Lind).
Equation 6-8 Page 204 (Lind).
Example Page 204.
What Is Probability
Distribution?
Ir. Muhril A., M.Sc., Ph.D.
21