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 σ-field 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(A|B) = 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(A|Bi)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∈*X*x*Y* g(x,y)f(x,y) | E[*g(x,y)*] = ∫x,y∈*X*x*Y* 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 bi-variate 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(y|x) = ∫-∞y f(x,v)/f(x) dv | f(y|x) = f(x,y)/f(x) where f(x) = ∫-∞∞ f(x,y) dy |

| Fy|x(y|x) = P(Y ≤ y|X = 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(y|x) = P(Y = y|X = x) |

| f(x,y) = f(x|y)f(y) or f(y|x)f(x) |

Conditional expectation | The conditional expectation of a RV Y, given X = x is E[Y|X =x] = ∑y∈Y yf(y|x) given that the conditional PMF is well-defined | E[h(X)g(Y)] = E[E[g(Y)|X]h(X)] = ∫(∫g(Y)f(Y|X) dx) h(X)f(x) dx |

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

| E[(aX + bY)|Z] = aE[X|Z] + bE[Y|Z] |

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

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