Discrete Probability Distributions

7/16/2019 Discrete Probability Distributions

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DISCRETE PROBABILITY DISTRIBUTIONS

Discrete

Uniform

Distribution

• The simplest discrete random variable is one that assumes only a

finite number of possible values, each with equal probability.

PMF

6-2



The Bernoulli sequence and the Binomial distribution

• In many engineering applications, there are often problems involving

the occurrence or non‐occurrence of an event, which is unpredictable,

in a sequence of discrete ‘trials’.

• For example, a piece of equipment may or may not malfunction over the duration of the project; in each year, the maximum flow of the

river may or may not exceed some specified flood level.

• Problems of this type may be modelled by a Bernoulli sequence,

which is based on the following assumptions.

• The experiment consists of n repeated trials.

• Each trial results in an outcome that may be classified as an occurrence or non‐

occurrence of an event.

• The

probability

of

success,

denoted

by

p,

remains

constant

from

trial

to

trial.• The repeated trials are independent.

6-3

The

Binomial

distribution

• The number X of successes in n Bernoulli trials is called a bi nomial

random variable. The probability distribution of this discrete random

variable is called the binomial distribution.

• A Bernoulli trial can result in a success with probability p and a failure

with Distribution probability q = 1 — p. Then the probability

distribution of the binomial random variable X , the number of

occurrences in n independent trials, is

Binomial PMF

)!(!

!

,...,2,1,0,);(

xn x

n

x

nwhere

n xq p x

nn, p x f xn x

6-4



The

Binomial

distribution

.

.

.

• The corresponding CDF is

• In spite of the simplicity, the Bernoulli model is quite useful in many

engineering applications. For example, in each of the following cases,

if the situation is repeated, the resulting series may be modelled as a

Bernoulli sequence

• The individual items produced on an assembly line may or may

not pass the inspection to ensure product quality

• In

an

earthquake

risk

zone,

a

building

may

or

may

not

be

damaged annually

n

k

xn xq p

x

nn, p xF

0

);(

6-5

• An industrial engineer is keenly interested in the "proportion

defective" in an industrial process. Often, quality control measures

and sampling schemes for processes are based on the binomial

distribution.

• The binomial distribution applies to any industrial situation where an

outcome of a process is dichotomous and the results of the process

are independent, with the probability of a success being constant

from trial to trial.

• The binomial distribution is also used extensively for medical

applications, where, a success or failure result is important. For

example, "cure" or "no cure" is important in pharmaceutical work.

The

Binomial

distribution

.

.

.

6-6



The

Binomial

distribution

.

.

.

6-7

• Each sample of water has a 10% chance of containing a particular

organic pollutant. Assume that the samples are independent with

regard to the presence of the pollutant. Find the probability that in the

next 18 samples, exactly 2 contain the pollutant.

Example

Determine the probability that at least four samples contain the pollutant

The

Binomial

distribution

.

.

.

6-8



• Determine the probability that 3 ≤X< 7

Example . . .

The

Binomial

distribution

.

.

.

6-9

The

Geometric

distribution

• In a Bernoulli sequence, the number of trials until a specified event

occurs for the first time is governed by the geometric distribution.

Note that the height of the line at x is (1-p) times

the height of the line at x- 1. That is, the

probabilities decrease in a geometric

progression. The distribution acquires its

name from this result.

6-10



The

Geometric

distribution

.

.

.

• In a time (or space) problem that is appropriately discretised into

corresponding intervals, and can be modelled as a Bernoulli

sequence, the number of time intervals until the first occurrence of

an

event

is

called

the

first

occurrence

time.• The probability distribution of the recurrence time is equal to that of

the first occurrence time. Therefore, the recurrence time in a

Bernoulli sequence is also governed by the geometric distribution.

• The mean recurrence time, which is popularly known in engineering

as the (average) return period is

1

21 1

...)321()1()( t

t

pqq p p pt T E T

6-11

The

Geometric

distribution

.

.

.

• A fixed offshore platform is designed for a wave height of 8m above the

mean sea level. This wave height corresponds to a 5% probability of being

exceeded per year.

(a) what is the probability that the platform will be subjected to the design

wave height within return period?(b) what is the probability that the first exceedance of the design wave

height will occur after the third year?

Example

6-12



The

Geometric

distribution

.

.

.

The return period of the design wave height is

By the geomteric distribution,

Example . . .

yearsT 2005.0

1

3585.095.01)20,8(1)20,8( 20

yr in H P yr in H P

8574.0])95.0(05.0)95.0(05.0)95.0(05.0[1)3(1)3( 131211

T PT P

6-13

The

Poisson

distribution

• Experiments yielding numerical values of a random variable X, the

number of outcomes occurring during a given time interval or in a

specified region, are called Poisson experiments.

• The given time interval may be of any length, such as a minute, a day,

a week, a month, or even a year.

• The specified region could be a line segment, an area, a volume, or

perhaps a piece of material.

• Examples: the number of

• telephone calls per hour received by an office

• days school is closed due to snow during the winter

• postponed games due to rain during a baseball season

• particles of contamination in semiconductor manufacturing

6-14



The

Poisson

distribution

.

.

.

• Properties of Poisson Process

• The number of outcomes occurring in one time interval or

specified region is independent of the number that occurs in any

other disjoint time interval or region of space. In this way we say

that the Poisson process has no memory.

• The probability that a single outcome will occur during a very

short time interval or in a small region is proportional to the

length of the time interval or the size of the region and does not

depend on the number of outcomes occurring outside this time

interval or region.

• The probability that more than one outcome will occur in such a

short time interval or fall in such a small region is negligible.• The number X of outcomes occurring during a Poisson experiment is

called a Poisson random variable, and its probability distribution is

called the Poisson distribution

6-15

The

Poisson

distribution

.

.

.

• It should be evident from the three principles of the Poisson process

that the Poisson distribution relates to the binomial distribution.

• Bernoulli sequence divides the time or space into appropriate

small intervals, and assumes that an event will either occur or not

occur (only two possibilities) within each interval, thus

constituting a Bernoulli trial

• However, if the event can randomly occur at any instant of time

(or at any point in space), it may occur more than once in any

given interval, in such cases, the occurrences of the event may be

more appropriately modelled with a Poisson process or Poisson

sequence

• The Bernoulli sequence approaches the Poison process as the time

(or space) interval is decreased

6-16



The

Poisson

distribution

.

.

.

• If Xt is number of occurrences in a time (or space) interval (0, t ), the

Poisson PMF is

where is the mean occurrence rate, i.e., the average number of

occurrences per unit time, distance.

• The mean number of occurrences in t is E(Xt)= t, and the variance is

also same.

• It is important to use consistent units in the calculation of

probabilities, means, and variances

,...2,1,0!

)( xe x

t x f t

x

6-17

The

Poisson

distribution

.

.

.

• Historical records of severe rainstorms in a town over the last 20

years indicated that there had been an average number of four

rainstorms per year. Assuming that the occurrences of rainstorms

may be modelled with Poisson process

a) what is the probability that there would not be any rainstorms next

year ?

b) what is the probability of four rainstorms next year?

c) what is the probability of two or more rainstorms in the next year?

d) generate a Poisson PMF for up to 12 rainstorms in the next year

Example

6-18



The

Poisson

distribution

.

.

.

Example

018.0!0

14)0( 14

0

e X f t

195.0!4

14)4( 14

4

e X f t

908.0!

141

!

14)2( 14

1

0

14

2

e

xe

x X F

x

x

x

x

t

6-19

The

Poisson

distribution

.

.

.

• Both in the Bernoulli sequence and the Poisson process, the

occurrences of an event between trials (in case of Bernoulli model)

and between intervals (in the Poisson model) are statistically

independent

• However, the probability of occurrence of an event in a given trial

may depend on earlier trials, and thus could involve conditional

probabilities. If this conditional probability depends on the

immediately preceding trial (or interval), the resulting model is a

Markov chain (or Markov Process).

6-20

Discrete Probability Distributions

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