7/29/2019 MA2216 Summary

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Given an event A of a sample space S , denote P(A) as its probability.

Axiom 1. 0 ≤ P(A) ≤ 1

Axiom 2. P(S ) = 1

Axiom 3. If  A1, A2, · · · are mutually exclusive ( AiAj = ∅), then 

P

∞i=1

Ai

=

∞i=1

P(Ai)

Theorem 1 (Inclusion-Exclusion Principle).

P(A1 ∪ A2 ∪ · · · ∪ An) =nr=1

(−1)r+1

1≤i1≤···≤ir≤n

P(Ai1Ai2 · · ·Air )

Definition 1 (Conditional Probability). Probability of  B given  A is

P(B|A) =P(AB)

P(A)

1. P(AB) = P(A)P(B|A) = P(B)P(A|B)

2. P(A1A2 · · ·An) = P(A1)P(A2|A1)P(A3|A1A2) · · ·P(An|A1 · · ·An−1)

Theorem 2 (Law of Total Probability).

P(B) =

ni=1

P(Ai)P(B|Ai)

Theorem 3 (Bayes’ Rule).

P(Ak|B) =P(Ak)P(B|Ak)ni=1 P(Ai)P(B|Ak)

For two events,

1. P(B) = P(A)P(B|A) + P(AC )P(B|AC )

2. P(A|B) =P(A)P(B|A)

P(A)P(B|A) + P(AC )P(B|AC )

Definition 2 (Independence). Two events are independent if  P(AB) =P(A)P(B). Also, P(A

|B) = P(A) and P(B

|A) = P(B).

Definition 3 (Cumulative Distribution Function). F X(x) = P{X ≤ x}Definition 4 (Probability Density Function). f X(x) = P{X  = x}Definition 5 (Continuous Random Variable). P{X  = x} for all  x ∈ R.F X is a continuous function, with  f X(x) = F X (x)

Definition 6.

E(X ) =

x xf x(x) for Discrete RV  ∞

−∞xf X(x) dx for Continuous RV 

E(aX  + b) = aE(X ) + b

Definition 7 (Moment Generating Function). M X (t) = E[etX ]If there is a  δ > 0 such that  M (t) <

∞for every 

|t

|<

∞, then  E(X k) =

M (k)(0)

Definition 8 (Variance). σ2X = E[(X −EX )2] = EX 2−(EX )2 Var (aX +b) = a2Var (X ), V ar(X +Y  ) = V ar(X )+ V ar(Y  ) if  X, Y   are independent.

Theorem 4 (Tail Sum Formula). For non-negative integer-valued r.v. X,

E[X ] =∞k=1

P(X ≥ k) =∞k=0

P(X > k)

Definition 9 (Binomial Distribution). X  is the number of success that occur in  n independent Bernoulli trials, denoted as X ∼ B(n, p).

b(i; n, p) =n

i

 pi(1 − p)n−i

EX  = np, Var (X ) = npq

Definition 10 (Poisson Distribution). X  is the number of occurenoccuring in a specific interval such that the occurences are independbetween intervals, denote as X ∼ P (λ)

 p(i; λ) = e−λλi

i!

EX  = λ, Var (X ) = λ

Suppose X  ∼ B(n, p), n is large and p is small (< 0.1), then X  ≈ P with λ = np.

Definition 11 (Geometric Distribution). X  is the number of trials rquired until a success is obtained for a repeated Bernoulli Trial. Den

X ∼ Geom ( p). f X(n) = (1 − p)n−1 p

EX  = 1p

, Var (X ) = 1−pp2

Definition 12 (Negative Binomial Distribution). X  is the numberBernoulli trials performed until  r successes are obtained. Denote as X

N B(r, p).

f X (n) =n − 1

r − 1

 pr(1 −  p)n−r

EX  = rp

, Var (X ) = r(1−p)p2

Definition 13 (Hyper Geometric Distribution). X  is the number of whballs drawn without replacement when  n balls are drawn in an urn withballs, out of which  m are white. Denote X ∼ H (n,N,m).

f X (n) = mx

N −mn−x N n

EX  = nmN 

, Var (X ) = N −nN −1

· n · mN 

(1 − mN 

)

Definition 14 (Exponential Distribution). A good model for waiting tiscenarios, X ∼ Exp(λ) if 

f X(n) =

λe−λx, x > 0

0 elsewhere 

EX  = 1λ

, Var (X ) = 1λ2

Exponential and geometric distribution are memoryless i.e. P{X > st|X > t} = P{X > s}Definition 15 (Gamma Distribution). X ∼ Γ(α, λ) if 

f X(n) = λα

Γ(α)xα−1e−λx, x > 0

0 elsewhere 

The Gamma function is defined by 

Γ(α) =

 ∞0

xα−1e−x dx

EX  = αλ

, Var (X ) = αλ2

Note that  ∞0

xα−1eλx dx =Γ(α)

λα

Definition 16 (Normal Distribution). X ∼ N (µ, σ2) if 

n(x; µ, σ) =1√ 

2πσe−

(x−µ)2

2σ2

The Gamma function is defined by 

Γ(α) =

 ∞0

xα−1e−x dx

EX  = µ, Var (X ) = σ2

The standard normal r.v. is denoted by Z  ∼ N (0, 1). For any X 

N (µ, σ2), we haveX − µ

σ∼ N (0, 1)

Given X ∼ B(n, p) such that np(1 − p) ≥ 10, we have X ≈ N (np,npq)

Remark 1 (Continuity Correction). If  X ∼ B(n, p), then 

{X  = k} = {k − 0.5 < X < k + 0.5}{X ≥ k} = {X ≥ k − 0.5}{X ≤ k} = {X ≤ k + 0.5}