Probability and Stats in Computer Science: Formula Sheet -...

Cheat sheet for the Final Exam
Discrete Distributions
Binomial probability mass function P ( x ) =
n !
x !( n − x )!
p
x
(1 − p )
n - x
for x = 0 , 1 , ..., n ,
Geometric probability mass function P ( x ) = (1 − p )
x - 1
p for x = 1 , 2 , ...
Poisson probability mass function P ( x ) =
e - λ λ x
x !
for x = 0 , 1 , ...
Continuous Distributions
Exponential density f ( x ) = λe
- λx
for 0 < x < ∞
Uniform density f ( x ) =
1
b − a
for a < x < b
Gamma density f ( x ) =
λ
r
Γ( r )
x
r - 1
e
- λx
for 0 < x < ∞
Gamma-Poisson formula P ( X < x ) = P ( Y ≥ r ) ; P ( X > x ) = P ( Y < r )
for X ∼ Gamma( r, λ ), Y ∼ Poisson( λx )
Normal density f ( x ) =
1
σ
√
2 π
e
- ( x - μ ) 2 / 2 σ 2
for −∞ < x < ∞
Normal approximation Binomial( n, p ) ≈ Normal
(
μ = np, σ =
√
np (1 − p )
)
for n ≥ 30, 0 . 05 ≤ p ≤ 0 . 95
Central Limit Theorem
( X
1
+ . . . + X
n
) − nμ
σ
√
n
→ Normal(0,1) as n → ∞
Expected values and variances
Distribution Bernoulli Binomial Geometric Poisson Exponential Gamma Uniform Normal
( p ) ( n, p ) ( p ) ( λ ) ( λ ) ( r, λ ) ( a, b ) ( μ, σ )
E ( X ) p np
1
p
λ
1
λ
r
λ
a + b
2
μ
Var( X ) p (1 − p ) np (1 − p )
1 − p
p 2
λ
1
λ 2
r
λ 2
( b − a )
2
12
σ
2
Stochastic processes
Binomial process: p = λ ∆, number of frames n =
t
∆
; number of events in time t is X ( t ) ∼ Binomial( n, p );
interarrival time is T = Y ∆, where Y ∼ Geometric( p )
Poisson process: number of events in time t is X ( t ) ∼ Poisson( λt ); interarrival time is T ∼ Exponential( λ )
Markov chains
k -step transition probability matrix P
k
= P
k
Steady state distribution is a solution of
{
πP = π
∑
π
i
= 1

Queuing Systems
Bernoulli single-server queuing system with capacity C
Arrival probability p
A
= λ
A
∆, departure probability p
S
= λ
S
∆.
Transition probabilities: p
00
= 1 − p
A
, p
01
= p
A
;
p
k,k - 1
= p
S
(1 − p
A
) , p
k,k
= p
A
p
S
+ (1 − p
A
)(1 − p
S
) , p
k,k +1
= p
A
(1 − p
S
) for 1 ≤ k ≤ C − 1;
p
C,C - 1
= p
S
(1 − p
A
) , p