special prob distribution

Special Probability
Distributions
Chapter 8

Ibrahim Bohari [e-mail] LOGO

Binomial Distribution

Binomial Distribution
In an experiment of n independent trials, where p is a the probability of a successful outcome q=1-p is the probability that the outcome is a failure
If X is a random variable denoting the number of successful outcome, the probability function of X is given P X  r  nCr p r q nr

Where q=1-p r=0,1,2,3,…..

X~B(n,p)
The n trials and probability of success p are called the parameters of the binomial distribution Binomial distribution consists of a number of successive trials of a random experiment. Each trial has only 2 outcome (success & failure)

Mean and Variance of the Binomial Distribution
The probability distribution of the Bernoulli trial with random variable X is given by Table 1
X=x
P(X=x)

0
1-p

1 p The expectation and variance can be calculated as follow E  X   01  p   1 p 
p

Mean and Variance of the Binomial Distribution
The expectation and variance can be calculated as follow Var  X   0 1  p   1  p   p 2
2

 p  p2
 p1  p 
 pq

2

Mean and Variance of the Binomial Distribution
Let X1, X 2  .........  X n be the random variables in n independent observations of the Bernoulli trial

Y  X1 , X 2  .........  X n
Where Y has the binomial distribution Y~B(n,p)
E Y   E  X 1  X 2  .........  X n 

 E  X 1   E  X 2   ......  E  X n 
 p  p  .......  p

 E Y   np

Mean and Variance of the Binomial Distribution
Var Y   Var  X 1  X 2  .........  X n 

 Var  X 1   Var  X 2   ......  Var  X n 
 pq  pq  .......  pq

Var Y   npq



X ~ N  , 2



Question 1
The random Variable, X has the binomial distribution X~B(6,0.2). Find the following probabilities: a)P(X=2)

Ans:
a) 0.2458

b) P X  5

b)0.0016

c)0.6554

c) P X  2

Question 2
The random Variable, X has the binomial distribution X~B(6,0.2). Find the following probabilities: a) P X  4

Ans:
a) 0.9984

b) P1  X  4

b) 0.7209

Question 3
The random Variable, X has the binomial distribution X~B(5,1/4). Find the following probabilities: a) P(X=2)

b) P X  3
c) P X  4

Ans:
a)0.2637

b)63/64

c)1/1024

Question 4
The random Variable, X has the binomial distribution X~B(5,1/4). Find the following probabilities: a) P X  3

b) P1  X  4
c) P2  X  5

Ans:
a) 53/512

b) 45/128

c)105/1024

Question 5
The random Variable, X has the binomial distribution X~B(5,1/4). Find the following probabilities: a) P3  X  5

Ans:
a) 1/64

b) P2  X  5

b)47/128

Question 6
A survey indicates that 20% of student exercise everyday. A random group of 10 student is chosen.
Find the probability that:
a) Exactly 4 students exercise everyday
b) At most 3 students exercise everyday

Ans:
a) 0.0881

b) 0.8791

Question 7
A survey indicates that 20% of student exercise everyday. A random group of 10 student is chosen.
Find the probability that:
a) At least 2 students exercise everyday
b) 1 to 5 students exercise everyday

Ans:
a)0.6242

b)0.8862

Question 8
A car manufacturer claims that only 5% of his raw materials at shopfloor are defective. If 60 raw materials are randomly selected, find the probability that: a) exactly 6 raw material are defective

Ans:
0.0490

Question 9
The manager of a bookstore claims that 80%books in their warehouse are academics books. If 50 books are randomly selected from the warehouse, find the probability that:
a) exactly 40 books are academic books

Ans:
0.1398

Question 10
A Calculus quiz consists of 50 multiple-choice questions. There are 4 choices for each question and only one choice is correct. A student makes a random guess for each question. Find the probability that the student obtained:

a) More than 20 correct questions
b) Exactly 2 correct questions

Ans: a)0.0063

b) 0.0001

Question 10(A)
The probability that a student from a class 5 Alpha supports Mr N. Karuppan to be elected as JPP of the class is 0.8. A sample of students is randomly . Find the probability that
a) Exactly 3 students support Mr N. Karuppan
b) More than 3 students support Mr N. Karuppan

Ans:
a) 0.2048

b) 0.7373

Question 11
In a shooting competition, the probability that a shooter hits the target is 0.6 . If the shooter makes 6 independent attempts, find the probability that he hits the target less than 3 times

Ans:
0.1792

Question 12
55% of the student in a school come to school by bus. A sample of 15 students is randomly chosen, find the probability that
a) Exactly 6 student come to school by bus

Ans:
0.1048

Question 13
The probability that an apple selected randomly from a bag is good 0.8. A sample of 10 apples is chosen, find the expectation and standard deviation of the number of goods apples

Ans:
E(X)=8;SD=1.265

Question 14
2% of the electric components made by manufacturing factory are defective. If a sample of 20 electric components are chosen, find the number of components expected to be defective and calculate the standard deviation

Ans:
E(X)=0.4; SD=0.6261

Question 15
The expected number of pupils who fails the
Mathematics test is 2.4 and the variance is 1.92. find the probability that
a) Exactly 5 pupils fail the test
b) Less than 2 pupils pass the test

Ans:
a) 0.05315

b) 0.0000002

Question 16
Analysis of JPJ records reveals that 45% candidates passing a car road test. Find the probability that in a random sample of 8 candidates
a)At least 7 candidates will pass the test

Ans:
0.0181

Question 17
The probability that Ali wins a chess game is 0.4.
Given that he plays only 5 games a week, find the probability that in any week he wins
a) Exactly four games
b) At least two games

Ans:
a) 0.0768

b)0.6630

Question 18
A ball manufacturer claims that only 15% of his balls are defective. If Ali has 6 of these balls to play with at a ball game, find the probability that
a) Exactly two of the balls will be defective
b) At most four balls will be defective

Ans:
a) 0.1762

b)0.9996

Question 19
A multiple choice test consists of 30 questions. There are four choices for each question and only one choice is correct. A student makes a random guess for each question. Find the probability that the student obtained
a) No correct answer
b) More than 16 correct answer

Ans:
a) 0.0002

b)0.0002

Question 20
It is known that 20% of the oranges produced by an orchard are rotten. If 10 oranges are chosen at random, find the probability that
a) Exactly 2 of them are rotten
b) At most 5 of them are rotten

Ans:
a)0.302

b)0.9936

Question 21
75% of the population of a certain town owns a car.
If a random sample of 50 people taken, find the probability that at least 35 people own a car.

Ans:
0.8369

Question 22
In an orchid farm, 3/5 of the orchids will produce flowers. Find the probability that if Puan Rosmah purchases 40 plants at least 20 will produce flowers.

Ans:
0.9256

Question 23
A new surgical procedure is successful 85% of the time. Then procedure is performed 8 times and it can be assumed that it is independent each time. Find the expected value and variance of the successful operations. Ans:
E(X)=6.8; Var(X)=1.02

Question 24
The manager of a carbonated drink company claims that 20% of the population drinks his product. If 10 people are randomly selected from the population, find the mean and variance of the number of people drinking his product

Ans:
2; 1.6

Poisson Distribution

Poisson Distribution
The Poisson distribution is used to describe random variables that count the number of occurrences in a particular time interval or space
Examples of events that might follow a Poisson distribution include
a) The number of telephone calls per day
b) The number of patients arriving at a clinic in the first hour it is opened
c) The number of misprint per page
d) The number of flaws in a metre of material
e) The number of parasites on a fish caught in a lake Poisson Distribution
X~ P0   where  is the parameter of the distribution


e 
P X  r   r! r

where r=0,1,2,…….

 is the mean number of occurrences in a given

interval follow a Poisson distribution. The variance is also 
Mean,

  E X   

Variance,   Var  X   
2

Poisson Distribution
Question 1
If X~Po(3), find the following probabilities:
a) P(X=2)

Ans:
a) 0.2240

b) P X  3

b) 0.5768

Question 2
If X~Po(3), find the following probabilities:
a) P X  3

Ans:
a)0.6472

b) P X  2

b) 0.1991

Question 3
Mr N. Karuppan expected to receive 6 telephone calls in for each hour. Find the probability that:

a) 4 calls received per hour
b) More than 2 calls received per hour
c) Less than 3 calls are received per 30 minutes

Ans:
a) 0.1339

b) 0.9380

c) 0.4232

Question 4
Mr Ganesh expected to receive 6 telephone calls in for each hour. Find the probability that:
a) Between 2 and 4 calls received per 20 minutes
b) Between 2 and 5 calls received per 10 minutes

Ans:
a)0.5413

b)0.0766

Question 5
A department store sells an average 3 electrical parts per day. Find the probability that on a given day that store will sell:
a) Exactly 4 electrical parts
b) At most 5 electrical parts
c) Three to eight electrical parts

Ans:
a)0.1681

b) 0.9161

c) 0.5730

Question 6
A department store sells an average 3 electrical parts per day. Find the probability that on a given day that store will sell per week:
a) Exactly 6 electrical parts
b) At least 15 electrical parts
c) More than 8 electrical parts

Ans:
a)0.0001

b)0.9284

c)0.9989

Question 7
The number of vehicle breakdowns on a highway per day is randomly distributed with a mean of 2. Find the probability that:
a) Exactly 1 vehicle break down
b) Not more than 4 vehicle break down
c) Between 2 and 5 vehicle break down for this week

Ans:
a)0.2707

b)0.9473

c)0.0017

Question 8
The mean number of students absent per day is two.
Find the probability that :
a) 5 students are absent
b) At least 3 students are absent

Ans:
a) 0.0361

b) 0.3233

Question 9
Operator expected received 15 phone calls per hour.
Find the probability of receiving
a) Exactly10 calls per hour
b) 12 or more calls per hour

Ans:
a) 0.04861

b)0.8152

Question 10
Salmi expected to receive 4 emails in a week. Find the probability of receiving :
a) No email this week
b) 3 emails at the most this week
c) 8 emails for the next 2 weeks

Ans:
a)0.0183

b)0.4335

c) 0.1395

Question 11
The mean number of motorcyles owned by each family is two. Find the probability that a family has
a) No motorcycle
b) At least one motorcycle
c) At most two motorcycle

Ans:
a) 0.1353

b) 0.8647

c)0.6767

Question 12
A rental car service center company has 5 cars available for rental each day. Assume that each car is rented out for the whole day and that the number of car rented out each day is randomly distributed with a mean of 2. Find the probability that the company cannot meet the demand of cars on any one day.

Ans:
0.0166

Question 13
The number of vehicle breakdowns on a highway in any one hour is a Poisson distribution with a mean of 2. An auto repair shop has only 1 tow truck to provide assistance. Assuming that each assistance will take at least one hour, find the probability the company cannot provide assistance to all the vehicle breakdowns in a particular hour.

Ans:
0.5940

Question 14
It is known that the probability of suffering side effects from a certain drug is 0.005. If 1500 persons use this drug, find the probability that
a) At most 1 person suffers from the side effects
b) 4 to 6 persons suffers from the side effects

Ans:
a)0.0047

b) 0.3191

Question 15
A certain facial treatment causes side effects for one in 2000 clients. If 3000 clients are given the treatment, find the probability that at most six clients will suffer side effects.

Ans:
0.9991

Question 16
The probability that Liza will have an accident at night is 0.003. What is the probability that Liza will have an accident more than 3 times in the next 4 years. Ans:
0.6406

Question 17
500 eggs were placed in a box. On an average, 0.4% of eggs would have broken before arriving at a grocery shop. Find the probability exactly 2 eggs were broken

Ans:
0.2707

Normal Distribution

Normal Distribution
The normal distribution is also used as an excellence approximation to distributions of discrete variables such as
a) Marks of examination
b) IQ scores
c) Profits of business ventures

Continuous Probability Distributions
Normal Distribution
Any normal curve with mean  and standard deviation  can be transformed to a normal curve with mean 0 and variance 1

Z ~ N 0,1
Z

X 



Normal Distribution



X ~ N ,

2



With mean,  and standard deviation, 

E X   
Var  X    2

NORMAL DISTRIBUTIONS
Question 1
If X~N(50,100), find the following probabilities:
a) P(X>60)

Ans:
a)0.1587

b)P(X