Transcript Chi-Squared Distribution - Erwin Sitompul

Probability and Statistics
Lecture 9
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
2 0 1 3
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Erwin Sitompul
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Chapter 6.5
Normal Approximation to the Binomial
Normal Approximation to the Binomial
 The probabilities associated with binomial experiments are readily
obtainable from the formula b(x;n, p) of the binomial distribution
or from the table when n is small.
 For large n, making the distribution table is not practical anymore.
 Nevertheless, the binomial distribution can be nicely approximated
by the normal distribution under certain circumstances.
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Erwin Sitompul
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Chapter 6.5
Normal Approximation to the Binomial
Normal Approximation to the Binomial
 If X is a binomial random variable with mean μ = np and variance
σ2 = npq, then the limiting form of the distribution of
Z
X  np
npq
as n  ∞, is the standard normal distribution n(z;0, 1).
 Normal approximation of b(x; 15, 0.4)
 Each value of b(x; 15, 0.4) is
approximated by P(x–0.5 < X < x+0.5)
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Chapter 6.5
Normal Approximation to the Binomial
Normal Approximation to the Binomial
P( X  4)  b(4;15, 0.4)
 15 C4 (0.4) 4 (0.6)11
 0.1268
P( X  4)  P(3.5  X  4.5)
 P(1.32  Z  0.79)
 0.1214
 Normal approximation of
9
b(4;15, 0.4) and
 b( x;15, 0.4)
x 7
P(7  X  9)   b( x;15, 0.4)
x 7
 0.3564
  np  (15)(0.4)  6
  npq  (15)(0.4)(0.6)  1.897
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P(7  X  9)  P(6.5  X  9.5)
 P(0.26  Z  1.85)
 0.3652
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Chapter 6.5
Normal Approximation to the Binomial
Normal Approximation to the Binomial
 The degree of accuracy, that is how well the normal curve fits the
binomial histogram, will increase as n increases.
 If the value of n is small and p is not very close to 1/2, normal
curve will not fit the histogram well, as shown below.
b( x; 6, 0.2)
b( x;15, 0.2)
 The approximation using normal curve will be excellent when n is
large or n is small with p reasonably close to 1/2.
 As rule of thumb, if both np and nq are greater than or equal to 5,
the approximation will be good.
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Chapter 6.5
Normal Approximation to the Binomial
Normal Approximation to the Binomial
 Let X be a binomial random variable with parameters n and p. For
large n, X has approximately a normal distribution with μ = np and
σ2 = npq = np(1–p) and
x
P( X  x)   b(k ; n, p)
k 0
 area under normal curve to the left of x  0.5
 P ( X  x  0.5)
( x  0.5)   

 PZ 




and the approximation will be good if np and nq = n(1–p) are
greater than or equal to 5.
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Chapter 6.5
Normal Approximation to the Binomial
Normal Approximation to the Binomial
The probability that a patient recovers from a rare blood disease is
0.4. If 100 people are known to have contracted this disease, what is
the probability that less than 30 survive?
  np  (100)(0.4)  40
n  100, p  0.4
29
P( X  30)   b( x;100, 0.4)
  npq  (100)(0.4)(0.6)  4.899
x 0
P( X  30)  P( X  29.5)
z
29.5  40
 2.143
4.899
 P ( Z  2.143)
 0.01608
 After interpolation
 1.608%
 Can you calculate the
exact solution?
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Chapter 6.5
Normal Approximation to the Binomial
Normal Approximation to the Binomial
A multiple-choice quiz has 200 questions each with 4 possible
answers of which only 1 is the correct answer. What is the probability
that sheer guess-work yields from 25 to 30 correct answers for 80 of
the 200 problems about which the student has no knowledge?
n  80, p 
1
4
  np  (80)( 14 )  20
  npq  (80)( 14 )( 34 )  3.873
z1 
P(25  X  30) 
30
 b( x;80,
x  25
1
4
24.5  20
30.5  20
 1.162, z2 
 2.711
3.873
3.873
)
 P(24.5  X  30.5)
 P(1.162  Z  2.711)
 P( Z  2.711)  P( Z  1.162)
 0.9966  0.8774
 0.1192
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Chapter 6.5
Normal Approximation to the Binomial
Normal Approximation to the Binomial
PU Physics entrance exam consists of 30 multiple-choice questions
each with 4 possible answers of which only 1 is the correct answer.
What is the probability that a prospective students will obtain
scholarship by correctly answering at least 80% of the questions just
by guessing?
n  30, p 
1
4
  np  (30)( 14 )  7.5
  npq  (30)( 14 )( 43 )  2.372
P( X  24) 
30
 b( x;30,
x  24
1
4
)
z
23.5  7.5
 6.745
2.372
 1  P( X  23.5)
 1  P( Z  6.745)
 0
 It is practically impossible to
get scholarship just by pure
luck in the entrance exam
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Chapter 6.6
Gamma and Exponential Distributions
Gamma and Exponential Distributions
 There are still numerous situations that the normal distribution
cannot cover. For such situations, different types of density
functions are required.
 Two such density functions are the gamma and exponential
distributions.
 Both distributions find applications in queuing theory and reliability
problems.
 The gamma function is defined by

( )   x 1e x dx
for α > 0.
0
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Chapter 6.6
Gamma and Exponential Distributions
Gamma and Exponential Distributions
 |Gamma Distribution| The continuous random variable X has a
gamma distribution, with parameters α and β, if its density
function is given by
 1
 1  x 
x
e
, x0
   ( )
f ( x)  

elsewhere
0,
where α > 0 and β > 0.
 |Exponential Distribution| The continuous random variable X
has an exponential distribution, with parameter β, if its density
function is given by
 1 x 
, x0
 e
f ( x)  

elsewhere
0,
where β > 0.
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Chapter 6.6
Gamma and Exponential Distributions
Gamma and Exponential Distributions
 Gamma distributions for certain values of
the parameters α and β
 The gamma distribution with α = 1 is called
the exponential distribution
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Chapter 6.6
Gamma and Exponential Distributions
Gamma and Exponential Distributions
 The mean and variance of the gamma distribution are
  
and
 2   2
 The mean and variance of the exponential distribution are

and
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2  2
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Chapter 6.7
Applications of the Gamma and Exponential Distributions
Applications of Gamma and Exponential Distributions
Suppose that a system contains a certain type of component whose
time in years to failure is given by T. The random variable T is
modeled nicely by the exponential distribution with mean time to
failure β = 5.
If 5 of these components are installed in different systems, what is
the probability that at least 2 are still functioning at the end of 8
years?

1
P(T  8)   et 5 dt
58
5
P( X  2)   b( x;5, 0.2)
x2
1
 1   b( x;5, 0.2)
 e 8 5  0.2
x 0
 1  0.7373
 The probability whether
the component is still
functioning at the end of 8
years
 0.2627
 The probability whether at
least 2 out of 5 such
component are still
functioning at the end of 8
years
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Chapter 6.7
Applications of the Gamma and Exponential Distributions
Applications of Gamma and Exponential Distributions
Suppose that telephone calls arriving at a particular switchboard
follow a Poisson process with an average of 5 calls coming per
minute.
What is the probability that up to a minute will elapse until 2 calls
have come in to the switchboard?
  1 5,   2
x
P( X  x )  
1
2

0
xe x  dx
 β is the mean time of the
event of calling
 α is the quantity of the
event of calling
1
P( X  1)  25 xe5 x dx  1  e5(1) (1  5)  0.96
0
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Chapter 6.7
Applications of the Gamma and Exponential Distributions
Applications of Gamma and Exponential Distributions
Based on extensive testing, it is determined that the average of time
Y before a washing machine requires a major repair is 4 years. This
time is known to be able to be modeled nicely using exponential
function. The machine is considered a bargain if it is unlikely to
require a major repair before the sixth year.
(a) Determine the probability that it can survive without major repair
until more than 6 years.
(b) What is the probability that a major repair occurs in the first
year?

(a)
1
P(Y  6)   et 4 dt  e6 4  0.223
46
(b)
1
P(Y  1)  1   e t 4 dt  1  e1 4  0.221
41
 Only 22.3% survives until
more than 6 years without
major reparation

1
 22.1% will need major
reparation after used for 1
year
1
  et 4 dt
40
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Chapter 6.8
Chi-Squared Distribution
Chi-Squared Distribution
 Another very important special case of the gamma distribution is
obtained by letting α = v/2 and β = 2, where v is a positive
integer.
 The result is called the chi-squared distribution, with a single
parameter v called the degrees of freedom.
 The chi-squared distribution plays a vital role in statistical
inference. It has considerable application in both methodology and
theory.
 Many chapters ahead of us will contain important applications of
this distribution.
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Chapter 6.8
Chi-Squared Distribution
Chi-Squared Distribution
 |Chi-Squared Distribution| The continuous random variable X
has a chi-squared distribution, with v degrees of freedom, if its
density function is given by
1

v 2 1  x 
x
e
, x0
 2v 2 (v 2)
f ( x)  

elsewhere
0,
where v is a positive integer.
 The mean and variance of the chi-squared distribution are
and
 v
 2  2v
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Chapter 6.9
Lognormal Distribution
Lognormal Distribution
 The lognormal distribution is used for a wide variety of
applications.
 The distribution applies in cases where a natural log
transformation results in a normal distribution.
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Chapter 6.9
Lognormal Distribution
Lognormal Distribution
 |Lognormal Distribution| The continuous random variable X has
a lognormal distribution if the random variable Y = ln(X) has a
normal distribution with mean μ and standard deviation σ. The
resulting density function of X is
2
 1
 ln( x )   
 2 x e
f ( x)  

0,
(2 2 )
, x0
x0
 The mean and variance of the chi-squared distribution are
E( X )  e
  2 2
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Var( X )  e
2   2
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2
(e 1)
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Chapter 6.9
Lognormal Distribution
Lognormal Distribution
Concentration of pollutants produced by chemical plants historically
are known to exhibit behavior that resembles a log normal
distribution. This is important when one considers issues regarding
compliance to government regulations.
Suppose it is assumed that the concentration of a certain pollutant,
in parts per million, has a lognormal distribution with parameters μ =
3.2 and σ = 1.
What is the probability that the concentration exceeds 8 parts per
million?
P( X  8)  1  P( X  8)
 ln(8)  3.2 
P( X  8)  F 
 F (1.12)  0.1314

1


 F denotes the cumulative distribution
function of the standard normal distribution
 a. k. a. the area under the normal curve
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Probability and Statistics
Homework 8A
1. (a) Suppose that a sample of 1600 tires of the same type are obtained at
random from an ongoing production process in which 8% of all such tires
produced are defective. What is the probability that in such sample 150
or fewer tires will be defective?
(Sou18. CD6-13)
(b) If 10% of men are bald, what is the probability that more than 100 in
a random sample of 818 men are bald?
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