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Collaborative Statistics: Formulas

Module by: Susan Dean, Dr. Barbara Illowsky

Summary: This module provides an overview of Statistics Formulas used as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

formula 1: Factorial

n ! = n ( n − 1 ) ( n − 2 ) . . . ( 1 ) n!=n(n−1)(n−2)...(1)

0 ! = 1 0!=1

formula 2: Combinations

n r = n ! ( n − r ) ! r ! n r = n ! ( n − r ) ! r !

formula 3: Binomial Distribution

X ~ B ( n , p ) X~B(n,p)

P ( X = x ) = n x p x q n − x P(X=x)= n x p x q n − x , for x = 0 , 1 , 2 , . . . , n x=0,1,2,...,n

formula 4: Geometric Distribution

X ~ G ( p ) X~G(p)

P ( X = x ) = q x − 1 p P(X=x)= q x − 1 p , for x = 1 , 2 , 3 , . . . x=1,2,3,...

formula 5: Hypergeometric Distribution

X~H ( r , b , n ) X~H ( r , b , n )

P ( X = x ) = ( ( r x ) ( b n − x ) ( r + b n ) ) P(X=x)= ( ( r x ) ( b n − x ) ( r + b n ) )

formula 6: Poisson Distribution

X~P ( μ ) X~P ( μ )

P ( X = x ) = μ x e − μ x ! P(X=x)= μ x e − μ x !

formula 7: Uniform Distribution

X ~ U ( a , b ) X~U(a,b)

f ( X ) = 1 b − a f(X)= 1 b − a , a < x < b a<x<b

formula 8: Exponential Distribution

X ~ Exp ( m ) X~Exp(m)

f ( x ) = m e − mx f(x)=m e − mx , m > 0 , x ≥ 0 m>0, x ≥ 0

formula 9: Normal Distribution

X ~ N ( μ , σ 2 ) X~N(μ, σ 2 )

f ( x ) = 1 σ 2 π e − ( x − μ ) 2 2 σ 2 f(x)= 1 σ 2 π e − ( x − μ ) 2 2 σ 2 , -∞<x <∞ -∞<x<∞

formula 10: Gamma Function

Γ ( z ) = ∫ 0 ∞ x z − 1 e − x dx Γ(z)= ∫ 0 ∞ x z − 1 e − x dx z>0z>0

Γ ( 1 2 ) = π Γ( 1 2 )= π

Γ ( m + 1 ) = m ! Γ(m+1)=m! for mm, a nonnegative integer

otherwise: Γ ( a + 1 ) = aΓ ( a ) Γ(a+1)=aΓ(a)

formula 11: Student-t Distribution

X ~ t df X~ t df

f ( x ) = ( 1 + x 2 n ) − ( n + 1 ) 2 Γ ( n + 1 2 ) nπ Γ ( n 2 ) f(x)= ( 1 + x 2 n ) − ( n + 1 ) 2 Γ ( n + 1 2 ) nπ Γ ( n 2 )

X = Z Y n X= Z Y n

Z ~ N ( 0,1 ) Z~N(0,1) , Y ~ Χ df 2 Y~ Χ df 2 ,nn = degrees of freedom

formula 12: Chi-Square Distribution

X ~ Χ df 2 X~ Χ df 2

f ( x ) = x n − 2 2 e − x 2 2 n 2 Γ ( n 2 ) f(x)= x n − 2 2 e − x 2 2 n 2 Γ ( n 2 ) , x>0x>0 , nn = positive integer and degrees of freedom

formula 13: F Distribution

X ~ F df ( n ) , df ( d ) X~ F df ( n ) , df ( d )

df ( n ) = df(n)=degrees of freedom for the numerator

df ( d ) = df(d)=degrees of freedom for the denominator

f ( x ) = Γ ( u + v 2 ) Γ ( u 2 ) Γ ( v 2 ) ( u v ) u 2 x ( u 2 − 1 ) [ 1 + ( u v ) x − . 5 ( u + v ) ] f(x)= Γ ( u + v 2 ) Γ ( u 2 ) Γ ( v 2 ) ( u v ) u 2 x ( u 2 − 1 ) [1+ ( u v ) x − . 5 ( u + v ) ]

X = Y u W v X= Y u W v , YY, WW are chi-square

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