Probability Distributions Summary - Exam P

Summary of Special Probability Distributions Continuous Probability Distributions Distribution

Parameters a, b

Uniform on [a, b]

µ, σ

Normal

λ

Exponential

λ, r

Gamma

n

Chi-square

Density Function 1 f ( x) = , a≤ x≤b b−a

f ( x) =

1 e σ 2π

( x − µ )2 − 2σ 2

Mean µ=

f ( x) =

Γ (r )

b+a . 2

µ=

x r −1e − λ x , x > 0

µ=

n −1 1 f ( x ) = n 2 n x 2 e− x 2 , x > 0 2 Γ( 2 )

(b − a )

σ2 =

µ

, −∞ < x < ∞

f ( x ) = λ e − λ x if x ≥ 0

λr

Variance

12

2

Moment Generating Function etb − eta , t ≠ 0. M (t ) = (b − a ) t

M (t ) = e

σ2 1

σ2 =

λ r

σ2 =

λ

µ=n

1

λ

M X (t ) =

.

λ

, t<λ λ −t r ⎛ λ ⎞ M (t ) = ⎜ ⎟ , t<λ ⎝ λ −t ⎠

2

r

λ2

σ 2 = 2n

1 2

µ t + σ 2t 2

⎛ 1 ⎞ M (t ) = ⎜ 1 2 ⎟ ⎝ 2 −t ⎠

n2

=

1

(1 − 2t )

n2

, t<

1 2

Discrete Probability Distributions Distribution Binomial Negative Binomial Geometric Poisson

Parameters n, p k, p

p

λ

Mass Function ⎛n⎞ n− x f ( x ) = ⎜ ⎟ p x (1 − p ) , x = 0, 1, 2, …, n ⎝ x⎠ ⎛ x − 1⎞ k x−k f ( x) = ⎜ ⎟ p (1 − p ) , ⎝ k − 1⎠ x = k , k + 1, k + 2, … .

f ( x ) = p (1 − p ) f ( x ) = e− λ

λx x!

x −1

, x = 1, 2, 3, …

, x = 0, 1, 2, …

Mean

Variance

µ = np

σ 2 = np (1 − p )

k µ= p

k (1 − p ) σ = p2

µ=

1 p

µ =λ

2

σ2 =

1− p p2

σ2 =λ

Moment Generating Function

(

M ( t ) = 1 + p ( et − 1)

M X (t ) =

( pe )

t k

⎡⎣1 − (1 − p ) et ⎤⎦

M X (t ) =

k

)

n

, t < − ln (1 − p )

pet , t < − ln (1 − p ) 1 − (1 − p ) et

M ( t ) = exp ⎡⎣λ ( et − 1) ⎤⎦