Download Common Discrete Distributions

Common Continuous Distributions
The Uniform Distribution
Definition: the Uniform Distribution has a constant (uniform) pdf from a to b.
1
Probability Function: f ( x)  Pr( X  x) 
ba
xa
Distribution Function: F ( x)  Pr( X  x) 
ba
ab
(also Median)
2
2
2


b  a
Length
var(
X
)


Variance:
12
12
bt
at
e e
MGF: M X (t ) 
t (b  a)
Moments: Mean: E[ X ] 
Hint: The probability of an interval event E=(c,d ) is proportional to its length.
d c
Pr( E ) 
ba
The Exponential Distribution
Probability Function: f ( x)  Pr( X  x) 
1

e
 1 x
(x > 0)
Distribution Function: F ( x)  Pr( X  x)  1  e
 1 x
Moments: Mean: E[X ]  
Variance: var( X )   2
Median: m   ln( 2)
1
1
t
MGF: M x (t ) 
1  t

Memoryless Property: PrX  x0  x X  x0   Pr X  x
The Pareto Distribution
Probability Function: f ( x)  Pr( X  x) 
 
(x > 0)
x    1

  
Distribution Function: F ( x)  Pr( X  x)  1  
 (x > 0)
 x  

Moments: Mean: E[ X ] 
(α> 1)
 1
Variance: var( X ) 
 2
(α> 2)
  12   2
The One Parameter Pareto Distribution
Probability Function: f ( y)  Pr(Y  y) 
 
y 1
(x > 0)

 
Distribution Function: F ( y )  Pr(Y  y )  1    (x > 0)
 y

Moments: Mean: E[Y ] 
(α> 1)
 1
Variance: var(Y ) 
 2
(α> 2)
  12   2
The Gamma Distribution
x
x 1e 
Probability Function: f ( x)  Pr( X  x)  
(x>0,α>0,θ>0)
  
For positive integer α Γ(α) = (α-1)!

Otherwise ( )   x 1e  x dx
0
x
x 2
x  1 







Distribution Function: F ( X )  Pr( X  x)  1  e 1  
 

1
!
2
!
(


1
)!


(x > 0,α integer)
Moments: Mean: E[ X ]  

x
Variance: var( X )   2
1
1
t
MGF: M x (t ) 


1  t 
Additive Property: If {Xi} is a fimily of independent, gamma distributed variables
with parameters αi and θ then  Xi is gamma distributed with parameters αi and
θ
Note: The exponential distribution is a gamma distribution with α = 1. Also the
sum of n exponential distributions with identical means of θ is a gamma
distribution with α = n and θ = θ.
The Chi-square Distribution (Parameter r = Degrees of Freedom)
r
Chi-square is a gamma distribution with   and   2
2
r
Probability Function: f ( x)  Pr( X  x) 
Moments: Mean: E[ X ]  r
1
x
x2 e 2
2 2  2r 
r
(x>0,r>0,θ>0)
Variance: var( X )  2r
1
MGF: M (t ) 
r
(1  2t ) 2
Additive Property: If {Xi} are independent chi-square distributed variables with ri
degrees of freedom, then Xi is shi-square distributed with  ri degrees of
freedom.
Note: The exponential distribution is a chi-square distribution with r = 2.
The Weibull Distribution

 x 
x 1e 
Probability Function: f ( x)  Pr( X  x) 
(x>0,τ>0,θ>0)

[0 ≤ x ≤ 1, a > 0, b > 0]
 1
Moments: Mean: E[X ]  1  
 
2
 2   2  1   1  
Variance: var( X ) 
2       
           
1
Median: m   ln( 2)  
The Beta Distribution
(a  b  1)! a1
x (1  x) b1
(a  1)! (b  1)!
[0 ≤ x ≤ 1, a > 0, b > 0]
(a  b) a1
Alternate Notation: f ( x) 
x (1  x) b1
(a)(b)
a
 a  a  1 
E[ X 2 ]  
Moments: Mean: E[ X ] 


ab
 a  b  a  b  1 
ab
Variance: var( X ) 
2
(a  b) (a  b  1)
Probability Function: f ( x)  Pr( X  x) 