chapter 3 discrete random variable.pdf

7/26/2019 Chapter 3 Discrete Random Variable.pdf

http://slidepdf.com/reader/full/chapter-3-discrete-random-variablepdf 1/23

Chapter 3: Discrete Random

Variable

- Binomial Probability Distribution

- Hypergeometry Distribution

- Poisson Distribution



3.1 Definition

A random variable is a variable whose value is a numerical

outcome of a random phenomenon.

As a real-valued function, random variable often describessome numerical quantity of a given event. For example, the

number of heads after a certain number of coin flips.

X variable

x possible value of the random variable

A random variable is called a discrete random variable if its

set of possible outcomes is countable.



Example:

Two balls are drawn in succession without replacement from

an urn containing 4 red balls and 3 black balls. Let Y denotes

the number of red balls, the values y are

Sample Space y

RR 2

RB 1

BR 1

BB 0



3.2 Probability Mass Function (pmf)

• A function that gives the probability that a discrete

random variable is exactly equal to some value.

• Also known as probability function or probability

distribution of the discrete random variable X

• Properties:

1) 0 12)

= = and

3) all

() = 1



Example 3.1

1) Find the value of k for a given probability distribution

function.

= for = 0,1,2,3,4Ans: Since

all

= 1

Therefore = 20.

2) Check whether the following can be defined as a

probability mass function. Explain your answer.

= + for = 1,2,3,4,5

Ans: No, since

all ≠ 1



3) A fair coin is tossed three times. Find the probability

distribution for the number of heads obtained.

Ans: Let X be the number of heads obtained

0 = = 0 = 12 × 1

2 × 12 =

18

1 = = 1 = 12 × 12 × 12 × 3 = 38 2 = = 2 = 1

2 × 12 × 1

2 × 3 = 38

3 = = 3 =

1

2 ×

1

2 ×

1

2 =

1

8Thus, the probability distribution of X is

0 1 2 3

() 1/8 3/8 3/8 1/8



3.3 Cumulative Distribution Function(CDF) of Discrete Random Variable

Cumulative distribution function(CDF)- of a discrete randomvariable with probability distribution function is

= = ≤

, ∞ < < ∞Example 3.2:

The probability distribution of X , the number of imperfections per 10

meters of a synthetic fabric in continuous rolls of uniform width, is given

by

Construct the cumulative distribution function of X.

0 1 2 3 4

()0.41 0.37 0.16 0.05 0.01



Ans:

Then, using (), finda) = 2b) > 1c)

( 3)d) ( < 2)e) (0 < < 3)f) (2 < 4)g) (1 3)



3.4 Expected Value of Discrete Random Variable

Definition:

For a discrete random variable X with probability distribution

(),

• the mean, or expected value of random variable X is

= = all

∙

•the mean, or expected value of random variable g is

() = all

() ∙



3.5 Variance of Discrete Random Variable

Variance, = = 2

=

2

= 2 =

where = , = =

Standard deviation, =



Example 3.3:

The random variable X, representing the number of

errors per 100 lines of software code, has the following

probability distribution:

Find the mean and variance of X .

Ans:

2 3 4 5 6

() 0.01 0.25 0.4 0.3 0.04



3.6 Binomial Distribution

The Binomial process possess all the following properties:

• The experiment consists of n repeated trials

•

Each trials results in an outcome that may be classifiedas a success or a failure.

• The probability of success, denoted by p, remains

constant from trial to trial.

• The repeated trials are independent.



Binomial Probability Distribution:

Notation: ~(, )The probability of obtaining successes from trials is given

by

= −

where = total number of trials

= probability of success

= 1 ; probability of failure

=num. of successes in

trials

For Binomial Distribution:

Mean, = Variance, =



Example 3.4:

1) The probability that a patient recovers from a delicateheart operation is 0.9. What is the probability that

exactly 5 of the next 7 patients having this operation

survive? Find the number of surviving patients that is

expected from this sample. (Ans: 0.1240)

2) It is known that 60% of mice inoculated with a serum

are protected from a certain disease. If 5 mice are

inoculated, find the probability thata) None contracts the disease; (Ans: 0.0778)

b) Fewer than 2 contract the disease; (Ans: 0.3370)

c) More than 3 contract the disease. (Ans: 0.0870)



3.7 Hypergeometric Distribution

• A discrete probability distribution that describes theprobability of x successes in n draws, without

replacement, from a finite population of size N

containing exactly k successes.

The Hypergeometric process possess all the following

properties:

• A random sample of size n is selected without

replacement from N items.• Of the N items, k may be classified as success and N-k

are classified as failure.



Hypergeometric Distribution

A sample of size n is selected from N items of which k are

labelled success and N-k labelled failure. The probabilityof the number of success obtained from the random

sample of size n is

=

, = 0,1,2, … ,

For Hypergeometric Distribution:

Mean, = Variance, = 1 −

−where

=



Example 3.5:

1) A homeowner plants 6 bulbs selected randomly from a

box containing 5 tulip bulbs and 4 daffodil bulbs. What is

the probability that he planted 2 daffodil bulbs and 4 tulip

bulbs? (Ans: 5/14)

2) If 6 of 18 new buildings in a city violate the building code,what is the probability that a building inspector, who

randomly selects 4 of the new buildings for inspection,

will catch

a) None of the buildings that violate the building code?b) 2 of the new building violate the building code?

c) At least 3 of the new buildings that violate the

building code?

(Ans: 0.1618; 0.3235; 0.0833)



Example 3.5:

3) A distributor buys 100 machine components from a localmanufacturer and 200 machine components from a

foreign manufacturer. If four components are selected

randomly and without replacement,

a) What is the probability that they are all from the localmanufacturer?

b) What is the probability that two or more components

in the sample are from the local manufacturer?

c) What is the probability that at least one component in

the sample is from the foreign manufacturer?

(Ans: 0.0119; 0.4075; 0.9881)



Example 3.5:

Multivariate Hypergeometric Distribution:4) A group of individuals is used for a biological case study.

The group contains 3 people with blood type O, 4 with

blood type A, and 3 with blood type B. What is the

probability that a random sample of 5 will contain 1person with blood type O, 2 with blood type A, and 2

with blood type B?

(Ans: 3/14)



3.8 Poisson Distribution

• A discrete probability distribution that expresses theprobability of a given number of events occurring in a

fixed interval of time and/or space if these events

occur with a known average rate and independently of

the time.

The Poisson process possess all the following properties:

• Occurrence in a given time interval is independent to

occurrence in other time intervals.• Probability of more than one success in given time

interval is negligible.



Poisson Distribution

The probability of a given number of outcomesoccurring in a given time interval or specified region

is given by

= −

! , = 0,1,2, … ,where is the average number of outcomes per unit

time, distance, area or volume.

For Poisson Distribution:

Mean, = Variance, =



Example 3.6:

1) If a bank receives on the average 6 bad cheques per

day, what are the probability that it will receive

a) 4 bad cheques on any given day?

b) 10 bad cheques over any 2 consecutive days?

(Ans: 0.1338; 0.1048)

2) At a checkout counter customers arrive at an

average of 1.5 per minute. Find the probability that

a) At most 4 will arrive in any given minute;

b) At least 3 will arrive during an interval of 2

minutes;

(Ans: 0.9814; 0.5768;)



Example 3.6:

3) In the inspection of paper produced by amachine, 0.2 imperfection is spotted per minute

on average. Find te probabilities of spotting

a) One imperfection in 3 minutes;

b) At least two imperfections in 5 minutes;

c) At most one imperfection in 15 minutes.

(Ans: 0.3293; 0.2642; 0.1991)

chapter 3 discrete random variable.pdf

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