Some Continuous Probability Distributions

Asmaa Yaseen

Review from Math 727

• Convergence of Random Variables

The almost sure convergence

The sequence converges to almost

surely denoted by , if

({ : ( ) ( )}) 1nP X X

.a sX Xn

XnX

Review from Math 727

• The convergence in Probability

The sequence converges to X in

probability denoted by , if

nXP

nX X

{ } 0lim P X Xnn

Review from Math 727 Quadratic Mean

Convergence

Almost Sure Convergence

Convergence in probability

Convergence in

Convergence in distribution

1L

Constant limit

Uniform integrability

Review from Math 727

• Let be a sequence of independent and identically distributed random variables, each having a mean and standard deviation . Define a new variable

Then, as , the sample mean X equals the

population mean of each variable

1 2, ,..., NX X X

1 2 ... nX X X

Xn

n

Review from Math 727

1 2

1

......(1)

1( ... )...(2)

...(3)

n

n

X X XX

n

X X Xnn

Xn

X

Review from Math 727

In addition 1

1

2

...var( ) var( )...(4)

var( ) var( ) ... var( )...(5)

var( )

n

n

X XX

nXX

Xn n

Xn

Review from Math 727

• Therefore, by the Chebyshev inequality, for all ,

As , it then follows that

0

2

2 2

var( )( )

XP X

n

n

lim ( ) 0nP X

Gamma, Chi-Squared ,Beta Distribution

Gamma Distribution The Gamma Function

for The continuous random variable X has a gamma

distribution, with parameters α and β, if its density function is given by

1

0

( ) xx e dx

0

( ; , )f x

11,

( )

x

x e

0X

0,Otherwise

0 0

Gamma, Chi-Squared ,Beta Distribution

Gamma’s Probability density function

Gamma, Chi-Squared ,Beta Distribution Gamma Cumulative distribution function

Gamma, Chi-Squared ,Beta Distribution

The mean and variance of the gamma distribution are :

2 2

Gamma, Chi-Squared ,Beta Distribution

The Chi- Squared Distribution The continuous random variable X has a chi-

squared distribution with v degree of freedom, if its density function is given by

( ; )f x v 2 1 2

2

1,

2 ( / 2)

v x

v x ev

0,x

0, Elsewhere ,

Gamma, Chi-Squared ,Beta Distribution

Gamma, Chi-Squared ,Beta Distribution

Gamma, Chi-Squared ,Beta Distribution

• The mean and variance of the chi-squared distribution are

Beta Distribution It an extension to the uniform distribution and

the continuous random variable X has a beta distribution with parameters and

2 2

v

v

0 0

Gamma, Chi-Squared ,Beta Distribution

If its density function is given by

( )f x

1 11(1 ) ,

( , )x x

0,

0 1,x

,elsewhere

The mean and variance of a beta distribution with parameters α and β are

22( ) ( 1)

and

Gamma, Chi-Squared ,Beta Distribution

Gamma, Chi-Squared ,Beta Distribution