uniform, gamma and beta pdf

8/8/2019 Uniform, Gamma and Beta PDF

1/15

The Uniform Distribution

0 ,

1

,

0 ,

x a

f x a x bb a

x b

! e e "

Uniform model: states that every interval of a fixed length has

the same probability

Random variablex is uniformly distributed over the interval from

a to b if it has the following probability density function :

ptan, ChE dept.


2/15

The variance is given by :

22 1

2

b

a

a b

x dxb aW

!

2( )

12

b a

!

The mean ofa uniformly distributed random variable is :

b

a

xdx

b aQ !

2a b

!

ptan, ChE dept.


3/15

Example :

A quality control engineer knows that the amounts of fill that a

machine puts into 12-ounce bottles are actually uniformly distributedbetween 11.90 and 12.40 ounces. Find the mean and the

percentage ofbottles containing more than 12.0 ounces.

The average fill per container is

then :

2

a bQ !

= (11.9 + 12.4) / 2

= 12.15 oz


4/15

ptan, ChE dept.

The percent of containers with more than 12.0 ounces is :

12.4

12.0

1( 12.0)

12.4 11.9P X dx" !

12.4

12.02 dx!

0.8 , or 80 %!

The standard deviation of fills is

2 2( ) (1 2 .4 11 .9 )0 .14 oz

1 2 1 2

b a ! ! !


5/15

The Gamma Distribution

Gamma function : an applied mathematics function defined as

an integral with limits from 0 to infinity

1

0

( ) , 0xx e dxEE Eg

+ ! "

Integration by parts technique yields :

1 2

0 0( ) ( 1) x xx e dx x e dxE EE E

g g + ! !

( 1) ( 1)E E! +

( 2 ) ? , (3) ? , ( 4 ) ? + ! + ! + !

recursive formula !

ptan, ChE dept.


6/15

at = 1 :1 1

0 0(1) 1x xx e dx e dx

g g + ! ! !

at =2 :

at =3 :

at = 4 :

ptan, ChE dept.

( 2 ) ( 2 1) ( 2 1) 1 (1) 1+ ! + ! + !

(3) (3 1) (3 1) 2 ( 2 ) 2 (1 ) 2 !+ ! + ! + ! !

( 4 ) ( 4 1) ( 4 1) 3 (3) 3( 2 !) 3 !+ ! + ! + ! !

Forany positive integer n , ( ) ( 1) !n n+ !

Using () = (1) (1)


7/15

Actual numerical values of the gamma function can be determined

from EXCEL.

Use EXCEL function : = EXP(GAMMALN( _ ))

Values of the gamma function , ()

() () () ()

ptan, ChE dept.


8/15

ptan, ChE dept.

Plot of the gamma function , ()

()


9/15

Gamma distribution : a probability density function with parameters

and defined as

1 /1( ) , 0 , 0 , 0

( )

xf x x e x

E F

EE F

F E

! " " "+

The mean of the gamma distribution is :

The variance is given by :2 2

W E F!

Q EF!

ptan, ChE dept.


10/15


11/15

Exponential distribution : special case of the gamma

distrib

ution with = 1.

The density function becomes :

1 /1( ) , 0 , 1 , 0 ( )

xf x x e xE F

EE F

F E ! " ! "

+

1 1 /

1

1, 0

(1)

xx e xF

F

! "

+

/1( ) , 0

xf x e x

F

F

! "The mean is : Q EF!

Q F!

ptan, ChE dept.


12/15

FE

EQ

!

The Beta Distribution

The beta distribution is defined on the unit interval. It has a probability

density function that is a function of the gamma function. The beta

distribution has the following probability density function :

1

1 1 , 0 1, 0, 0

0 , otherwise

x x xf x

FE

E FE FE F

+

" "! + +

The mean of the beta distribution is

ptan, ChE dept.


13/15

Example :

Consider the beta distribution when = 4 and when= 4.Verify that thebeta density is a density function by showing that it integrates to 1.

4 1 34 1 34 4 7!

1 14 4 3!3!

is the density unction.

f x x x x x+ ! ! !

+ +

Integrating, we find thatf(x) is a density since the integral equals 1.

The mean is found to be :

ptan, ChE dept.

0.5E

QE F

! !

3 3140 (1 )x x


14/15

ptan, ChE dept.

The beta density function f(x)= 140x3(1x3)


15/15

A civil engineer knows from experience that the proportion of highway

sections requiring repairs in any given year in Douglas county is arandom variable with a beta distribution having = 1 and = 3.

Determine the probability that at most 40% of the highway sections

will require repairs in any given year.

The density function is .10,132

! xxxf

!!4.0

0

27840.0134.0 dxxxP

ptan, ChEdept.

The probability is

Example :

uniform, gamma and beta pdf

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