Some Continuous Probability
Distributions
Asmaa Yaseen
Review from Math 727
• Convergence of Random Variables
The almost sure convergence
The sequence X n converges to X
denoted by X
n
a.s ,

X
if
P({ : X n ( )  X ( )})  1
almost surely
Review from Math 727
• The convergence in Probability
The sequence X n converges to X in probability
denoted
P
X

by n  X
, if
lim P{ X n  X  } 0
n 
Review from Math 727
Quadratic Mean
Convergence
Almost Sure
Convergence
1
Convergence in L
Convergence in
probability
Constant
limit
Convergence in
distribution
Uniform
integrability
Review from Math 727
• Let X1, X 2 ,..., X N
be a sequence of
independent and identically distributed
random variables, each having
a mean  and standard deviation  . Define
a new variable
X 1  X 2  ...  X n
X
n
Then, as n  , the sample mean X equals the
population mean  of each variable
Review from Math 727
X
X
X
X
X 1  X 2  ...  X n

...(1)
n
1
 ( X 1  ...  X n )...(2)
n
n

...(3)
n

Review from Math 727
In addition
X 1  ...  X n
var( X )  var(
)...(4)
n
Xn
X1
var( X )  var( )  ...  var( )...(5)
n
n
var( X ) 

2
n
Review from Math 727
• Therefore, by the Chebyshev inequality, for
all   0 ,
P( X     ) 
var( X )
2
2
 2
n
As n  , it then follows that
lim P( X     )  0
n 
Gamma, Chi-Squared ,Beta
Distribution
Gamma Distribution
The Gamma
Function

( )   x 1e x dx
for   0
0
The continuous random variable X has a gamma
distribution, with parameters α and β, if its density
function is given by
f ( x; ,  ) 
 0  0
x
1
 1

x
e
,

 ( )
0,
X 0
Otherwise
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 chisquared distribution with v degree of
freedom, if its density function is given by
1
f ( x; v) 
v
2 2 (v / 2)
0,
x
v
21
e
x
2
,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
 v
 2  2v
Beta Distribution
It an extension to the uniform distribution and
the continuous random variable X has a beta
distribution with parameters   0 and   0
Gamma, Chi-Squared ,Beta
Distribution
If its density function is given by
f ( x) 
1
 1
 1
x (1  x) ,
 ( ,  )
0  x  1,
elsewhere,
0,
The mean and variance of a beta distribution with
parameters α and β are


2

 
and


(   ) 2 (    1)
Gamma, Chi-Squared ,Beta
Distribution
Gamma, Chi-Squared ,Beta
Distribution