Sample Space 
The set of all possible outcomes of an experiment is called the sample space and is denoted by Ω. 
Sigma field 
A collection of sets F of Ω is called a σfield if it satisfies the following conditions: 

1. ∅ ∈ F 2. If A1,...,∈ F then U∞1 Ai ∈ F 

3. If A ∈ F then A^{c} ∈ F 
Probability 
A probability measure P on (Ω, F ) is a function P : F → [0, 1] which satisfies: 

1.P(Ω)=1 and P(∅)=0 
2. 
Conditional Probability 
Consider probability space (Ω, F , P) and let A, B ∈ F with P(B) > 0. Then the conditional probability that A occurs given B occurs is defined to be: P(AB) = P(A ∩ B) / P(B) 
Total Probability 
A family of sets B1, . ., Bn is called a partition of Ω if: ∀i !=j Bi ∩Bj =∅ and U∞1 Bi =Ω 
P(A) = ∑n1 P(ABi)P(Bi) 
P(A) = ∑n1 P(A∩Bi) 
Independence 
Consider probability space (Ω, F , P) and let A, B ∈ F . A and B are independent if P(A ∩ B) = P(A)P(B) 

More generally, a family of F−sets A1,...,An (∞ > n ≥ 2) are independent if P(∩n1 Ai) = ∏ n1 P(Ai) 
Random Variable (RV) 
A RV is a function X : Ω → R such that for each x ∈ R, {ω ∈ Ω : X(ω) ≤ x} ∈ F. Such a function is said to be F−measurable 
Distribution Function 
Distribution function of a random variable X is the function F : R → [0, 1] given by F(x)=P(X ≤x), x∈R. 
Discrete RV 
A RV is said to be discrete if it takes values in some countable subset X = {x1,x2,...} of R 
PMF 
PMF of a discrete RV X, is the function f :X→[0,1] defined by f(x)=P(X =x). It satisfy: 
PDF 
function f is called the probability density function (PDF) of the con tinuous random variable X 

1. set of x s.t. f(x) != 0 is countable 

f(x) = F'(x) 

2. ∑x∈X f(x) = 1 

F(x) = ∫∞x f(u) du 

3. f(x) ≥ 0 
Independence 
Discrete RV X and Y are indie if the events {X = x} & {Y =y} are indie for each(x,y)∈X×Y 
The RV X and Y are indie if {X≤x} {Y≤y} are indie events for each x, y ∈ R 

P(X,Y) = P(X=x)P(Y=y) 

f(x,y) = f(x)f(y) 
f(x,y) = f(x)f(y) F(x,y) v 

E[XY] = E[X]E[Y] 
Expectation 
expected value of RV X on X, 
The expectation of a continuous random variable X with PDF f is given by 

E[X] = ∑x∈X xf(x) 
E[X] = ∫x∈X xf(x) dx 

E[g(x)] = ∑x∈X g(x)f(x) 
E[g(x)] = ∫x∈X g(x)f(x) dx 
Variance 
spread of RV 
E[(X − E[X])^{2}] 
E[X^{2}]  E[X]^{2} 
MGF (uniquely characterises distribution) 
M(t) = E[e^{Xt}] = ∑x∈X e^{Xt} f(x) 
t∈T s.t. t for ∑x∈X e^{Xt} f(x) < ∞ 

M(t) = E[e^{Xt}] = ∫x∈X e^{Xt} f(x) dx 
t∈T s.t. t for ∫x∈X e^{Xt} f(x) dx < ∞ 

M(t1,t2) = E[e^{Xt1+Yt2}] = ∫z e^{Xt1+Yt2} f(x,y) dxdy (t1,t2)∈T 
E[X] = ∂/∂t1 M(t1,t2) t1=t2=0 
E[XY] = ∂^{2}/∂t1∂t2 M(t1,t2) t1=t2=0 

E[X^{k}] = M^{k}(0) 
Moment 
Given a discrete RV X on X, with PMF f and k ∈ Z^{+}, the k^{th} moment of X is 
E[X^{k}] 
Central Moment 
k^{th} central moment of X is 
E[(X − E[X])^{k}] 
Dependence 
Joint distribution function F : R^{2} → [0,1] of X,Y where X and Y are discrete random variables, is given by F(x,y) = P(X≤x∩Y≤y) 
The joint distribution function of X and Y is the function F : R2 → [0, 1] given by F(x,y)=P(X≤x,Y ≤y) 

Joint mass function f : R2 → [0, 1] is given by f(x,y) = P(x∩y) 
The random variables are jointly continuous with joint PDF f : R2 → [0, ∞) if F(x, y) = ∫∞y∫∞x f(u,v) dudv 


f(x,y) = ∂^{2}/∂x∂y F(x,y) 
Marginal 
f(x) = ∑y∈Y f(x,y) 
f(x) = ∫y∈Y f(x,y)dy 
F(x) = lim y>∞ F(x,y) F(x) = ∫∞x∫∞∞ f(u,y) dydu 

E[g(x,y)] = ∑x,y∈XxY g(x,y)f(x,y) 
E[g(x,y)] = ∫x,y∈XxY g(x,y)f(x,y) dxdy 
Covariance 
indie => E[XY] = E[X]E[Y], Cov = 0 => ρ = 0 
ρ = 0 => E[XY] = E[X]E[Y] 

Cov[X,Y] = E[(X − E[X])(Y − E[Y])] 
Cov[X,Y] = E[XY]  E[X]E[Y] 
Correlation 
Gives linear relationship (+/). ρ close to 1 is strong, close to 0 is weak 
special for bivariate normal, indie <=> uncorrelated 

ρ(X,Y)= Cov[X,Y] / sqrt(Var[X]Var[Y]) 
Conditional distribution 
The conditional distribution function of Y given X, written FY x(·x), is defined by 
F(yx) = ∫∞y f(x,v)/f(x) dv 
f(yx) = f(x,y)/f(x) where f(x) = ∫∞∞ f(x,y) dy 

Fyx(yx) = P(Y ≤ yX = x) 

for any x with P(X =x)>0. The conditional PMF of Y given X =x is defined by ... when x is s.t. P(X =x)>0 

f(yx) = P(Y = yX = x) 

f(x,y) = f(xy)f(y) or f(yx)f(x) 
Conditional expectation 
The conditional expectation of a RV Y, given X = x is E[YX =x] = ∑y∈Y yf(yx) given that the conditional PMF is welldefined 
E[h(X)g(Y)] = E[E[g(Y)X]h(X)] = ∫(∫g(Y)f(YX) dx) h(X)f(x) dx 

E[YX =x] = ∑y∈Y yf(yx) 
E[E[YX]] = E[Y] 
E[E[YX]g(X)] = E[Yg(X)] 

E[(aX + bY)Z] = aE[XZ] + bE[YZ] 

if X and Y are independent 
E[XY] = E[X] 
Var[XY] = E[X^{2}Y]  E[XY]^{2} 
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