Transcript Chapter 1 - UniMAP Portal

1.4 PROBABILITY DISTRIBUTIONS
Probability Distributions
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A probability function is a function which assigns probabilities
to the values of a random variable.
Individual probability values may be denoted by the symbol
P(X=x), in the discrete case, which indicates that the random
variable can have various specific values.
All the probabilities must be between 0 and 1;
0≤ P(X=x)≤ 1.
The sum of the probabilities of the outcomes must be 1.
∑ P(X=x)=1
It may also be denoted by the symbol f(x), in the continuous,
which indicates that a mathematical function is involved.
Probability
Distributions
Discrete
Probability
Distributions
Binomial
Poisson
Continuous
Probability
Distributions
Normal
Example 1.24
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Check whether the distribution is a probability distribution.
Solution
X
0
1
2
3
4
P(X=x)
0.125
0.375
0.025
0.375
0.125
4
 P( X  x)  P( X  0)  P( X  1)  P( X  2)  P( X  3)  P( X  4)
0
= 0.125+0.375+0.025+0.375+0.125
= 1.025
1
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# so the distribution is not a probability distribution.
Binomial Distribution
An experiment in which satisfied the following characteristic is
called a binomial experiment:
1. The random experiment consists of n identical trials.
2. Each trial can result in one of two outcomes, which we denote
by success, S or failure, F.
3. The trials are independent.
4. 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.
Examples:
1.
No. of getting a head in tossing a coin 10 times.
2.
No. of getting a six in tossing 7 dice.
3.
A firm bidding for contracts will either get a contract or not
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 )  nC x p x q n  x ;
x  0,1, 2....n
The Mean and Variance of X if X ~ B(n,p) are
Mean
Variance
  E ( X )  np
:
:
 2  V ( X )  np(1  p)  npq
Std Deviation :   npq
where n is the total number of trials, p is the probability of
success and q is the probability of failure.
Example 1.25
Given that X b(12,0.4), find
a) P( X  2)
b) P(2  X  5)
c) E( X )
d) Var( X )
Cumulative Binomial Distribution
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When the sample is relatively large, tables of Binomial are
often used. Since the probabilities provided in the tables are
in the cumulative form P  X  k  the following guidelines can
be used:
Example 1.26
In a Binomial Distribution, n =12 and p = 0.3. Find the following
probabilities.
a) P ( X  5) 
b) P ( X  5) 
c)
P( X  9) 
d)
P(5  X  9) 
e)
P (3  X  5) 
Example 1.26

In a Binomial Distribution, n =12 and p = 0.3. Find the
following probabilities.
a) P( X  5)  P( X  4)  0.7237
b) P( X  5)  0.8822
c) P( X  9)  1  P( X  8)  1  0.9983  0.0017
d) P(5  X  9)  P( X  9)  P( X  5)
 0.9998  0.8822  0.1176
e) P(3  X  5)  P( X  5)  P( X  2)
 0.8822  0.2528 
Exercise 1.5:
1. Let X be a Binomial random variable with parameters n = 20, p =
0.4. By using cumulative binomial distribution table, find :
a) P( X  14)
b) P( X  10)
c) P( X  13)
d) P( X  10)
e) P(9  X  15)
f) P(7  X  15)
2. Use cumulative binomial table for n = 5 and p = 0.6 to find the
probabilities below:
a) P( X  3)
ans : 0.6630
b) P( X  3)
ans : 0.6826
Example 1.27:
In a recent year, 10% of the luxury car sold were black. If 10 cars are
randomly selected, find the following probabilities :
a)
b)
c)
d)
At least five cars are black
At most three cars are black
Exactly four cars are black
Less than one are not black
Exercises 1.6 :
1. Consider a binomial random variable with n = 8 and p = 0.3. Fill in
table below:
k
P( X  k )
0
1
2
3
4
5
6
7
8
2. Fill in the blanks of table below:
Problem
Less than three
More than three
At most three
Fewer than three
Between 3 and 5
Exactly three
At least five
Write the
probability
Rewrite the
probability (if
needed)
Find the probabillity
Exercises 1.7 :
1. Suppose you will be attending 6 hockey games. If each game will
go to overtime with probability 0.10, find the probability that
a)
b)
At least 1 of the games will go to overtime.
At most 1 of the games will go to overtime.
2. Statistics indicate that alcohol is a factor in 50 percent of fatal
automobile accidents. Of the 3 fatal automobile accidents, find the
probability that alcohol is a factor in
a)
b)
Exactly two
At least 1
Poisson Distribution
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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
Examples:
1.
No. of cars passing a toll booth in one hour.
2.
No. defects in a square meter of fabric
3.
No. of network error experienced in a day.
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 
P( X  x) 
x!
x
for x  0,1, 2,3,...
A random variable X having a Poisson distribution can also be
written as
X
Po ( )
with E ( X )   and Var ( X )  
Example 1.28
Consider a Poisson random variable with   3 . Calculate the
following probabilities :
a) Write the distribution of Poisson
b) P( X  0) 
c) P( X  1) 
d) P( X  1) 
Exercises 1.8:
1. Consider a Poisson random variable with  .3Fill in table below:
k
P( X  k )
0
1
2
3
4
5
6
7
8
2. Fill in the blanks of table below:
Problem
Less than three
More than three
At most three
Fewer than three
Between 3 and 5
Exactly three
At least five
Write the
probability
Rewrite the
probability (if
needed)
Find the probabillity
Example 1.29
The average number of traffic accidents on a certain section of
highway is two per week. Assume that the number of accidents
follows a Poisson distribution with mean is 2.
i)
ii)
Find the probability of no accidents on this section of highway
during a 1-week period
Find the probability of a most three accidents on this section of
highway during a 2-week period.
Solution:
i) P( X  0) 
ii) P ( X  3) 
Exercise 1.9:
1. The demand for car rental by AMN Travel and Tours can be
modelled using Poisson distribution. It is known that on average 4
cars are being rented per day. Find the probability that is randomly
choosing day, the demand of car is:
a) Exactly two cars
b) More than three cars
2. Overflow of flood results in the closure of a causeway. From
past records, the road is closed for this reason on 10 days during
a 20-year period. At a village, the villagers were concern about the
closure of the causeway because the causeway provides the only
access to another village nearby.
a) Determine the probability that the road is closed less than 5
days in 20 years period.
b) Determine the probability that the road is closed between 2
and 6 days in five years period.
Poisson Approximation of Binomial Probabilities
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The Poisson distribution is suitable as an approximation of
Binomial probabilities when n is large and p is small.
Approximation can be made when n  30 , and either
np  5 or nq  5
Example 1.30:
Given that X ~ B(1000,0.004), find :
a) P( X  7)
b) P( X  9)
Exercise 1.10:
1.
Given that X ~ B(2, 0.4)
Find P( X  0), P( X  2), P( X  2), P( X  1), E ( X ),Var ( X ).
(ans: 0.36, 0.16, 1.0, 0.64, 0.8, 0.48).
2.
In Kuala Lumpur, 30% of workers take public transportation. In a
sample of 10 workers,
i) what is the probability that exactly three workers take public
transportation daily? (ans: 0.2668)
ii) what is the probability that at least three workers take public
transportation daily? (ans: 0.6172)
3. Let X ~ P0 (12). Using Poisson distribution table, find
i) P( X  8) and P( X  8) (ans: 0.1550, 0.0655)
ii) P( X  4) and P( X  4) (ans: 0.9977, 0.9924)
iii) P(4  X  14)
(ans: 0.7697)
4. Last month ABC company sold 1000 new watches. Past
experience indicates that the probability that a new watch will
need repair during its warranty period is 0.002. Compute the
probability that:
i) At least 5 watches will need to warranty work. (ans: 0.0527)
ii) At most 3 watches will need warranty work. (ans: 0.8571)
iii) Less than 7 watches will need warranty work. (ans: 0.9955)