Statistics tables and formulas

Formulae
Distribution Probability function Mean Variance MGF
Bernoulli (; ) =  (1 − )1−  (1 − )  + (1 − )
 = 0 1 ¡ ¢
Binomial (;  ) =   (1 − )−  (1 − ) [ + (1 − )]
 = 0 1  
( )( − ) 
¡ − ¢ ¡ ¢
Hypergeometric (;   ) =  −  1− 
does not exist
()  −1  
 = 0 1  ( ) in closed form
¡ ¢  (1−) (1−) 
Negative binomial (;  ) = +−1−1
 (1 − )  2 [1− (1−)]
 = 0 1 2 
1− 1− 
Geometric (; ) = (1 − )  2 1− (1−)
 = 0 1 2 
 −
Poisson (; ) =  ! 
  exp [ ( − 1)]
 = 0 1 2  h i
2
−2
Standard normal () = √12 exp 2
0 1 2
-∞    ∞ h i
−(−)2  2 2
Non-standard normal(;  ) = √1
exp  2 + 2
 2 22
-∞    ∞
1 + (−)2  −
Continuous uniform (;  ) = − 2 12 (−)
≤≤
−1 −
Standard gamma (; ) =  Γ()   (1 − )−
0
−1 −
Non-standard gamma (;  ) =    Γ()   2 (1 − )−
0
Exponential (; ) = − 1 12 (1 − )
≥0
2−1 −2
Chi-squared (; ) = 22 Γ(2)  2 (1 − 2)−2
≥0

Target parameter,  Point estimator, b
 (b
) Standard error, 
√
   
p 
 
 b =   q 2
1 2
1 − 2 1 − 2 1 − 2 + 22
q 1
1 1  
1 − 2 b1 − b2 1 − 2 1
+ 2 22


− b
= 
 ± 2 




1