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Sample Space
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The set of all possible outcomes of an experiment is called the sample space and is denoted by Ω.
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Sigma field
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A collection of sets F of Ω is called a σ-field if it satisfies the following conditions:
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1. ∅ ∈ F 2. If A1,...,∈ F then U∞1 Ai ∈ F
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3. If A ∈ F then Ac ∈ F
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Probability
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A probability measure P on (Ω, F ) is a function P : F → [0, 1] which satisfies:
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1.P(Ω)=1 and P(∅)=0
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2.
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Conditional Probability
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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)
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Total Probability
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A family of sets B1, . ., Bn is called a partition of Ω if: ∀i !=j Bi ∩Bj =∅ and U∞1 Bi =Ω
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P(A) = ∑n1 P(A|Bi)P(Bi)
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P(A) = ∑n1 P(A∩Bi)
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Independence
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Consider probability space (Ω, F , P) and let A, B ∈ F . A and B are independent if P(A ∩ B) = P(A)P(B)
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More generally, a family of F−sets A1,...,An (∞ > n ≥ 2) are independent if P(∩n1 Ai) = ∏ n1 P(Ai)
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Random Variable (RV)
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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
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Distribution Function
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Distribution function of a random variable X is the function F : R → [0, 1] given by F(x)=P(X ≤x), x∈R.
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Discrete RV
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A RV is said to be discrete if it takes values in some countable subset X = {x1,x2,...} of R
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PMF
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PMF of a discrete RV X, is the function f :X→[0,1] defined by f(x)=P(X =x). It satisfy:
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PDF
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function f is called the probability density function (PDF) of the con- tinuous random variable X
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1. set of x s.t. f(x) != 0 is countable
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f(x) = F'(x)
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2. ∑x∈X f(x) = 1
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F(x) = ∫-∞x f(u) du
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3. f(x) ≥ 0
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Independence
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Discrete RV X and Y are indie if the events {X = x} & {Y =y} are indie for each(x,y)∈X×Y
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The RV X and Y are indie if {X≤x} {Y≤y} are indie events for each x, y ∈ R
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P(X,Y) = P(X=x)P(Y=y)
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f(x,y) = f(x)f(y)
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f(x,y) = f(x)f(y) F(x,y) v
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E[XY] = E[X]E[Y]
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Expectation
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expected value of RV X on X,
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The expectation of a continuous random variable X with PDF f is given by
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E[X] = ∑x∈X xf(x)
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E[X] = ∫x∈X xf(x) dx
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E[g(x)] = ∑x∈X g(x)f(x)
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E[g(x)] = ∫x∈X g(x)f(x) dx
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Variance
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spread of RV
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E[(X − E[X])2]
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E[X2] - E[X]2
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MGF (uniquely characterises distribution)
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M(t) = E[eXt] = ∑x∈X eXt f(x)
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t∈T s.t. t for ∑x∈X eXt f(x) < ∞
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M(t) = E[eXt] = ∫x∈X eXt f(x) dx
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t∈T s.t. t for ∫x∈X eXt f(x) dx < ∞
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M(t1,t2) = E[eXt1+Yt2] = ∫z eXt1+Yt2 f(x,y) dxdy (t1,t2)∈T
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E[X] = ∂/∂t1 M(t1,t2) |t1=t2=0
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E[XY] = ∂2/∂t1∂t2 M(t1,t2) |t1=t2=0
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E[Xk] = Mk(0)
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Moment
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Given a discrete RV X on X, with PMF f and k ∈ Z+, the kth moment of X is
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E[Xk]
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Central Moment
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kth central moment of X is
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E[(X − E[X])k]
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Dependence
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Joint distribution function F : R2 → [0,1] of X,Y where X and Y are discrete random variables, is given by F(x,y) = P(X≤x∩Y≤y)
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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)
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Joint mass function f : R2 → [0, 1] is given by f(x,y) = P(x∩y)
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The random variables are jointly continuous with joint PDF f : R2 → [0, ∞) if F(x, y) = ∫-∞y∫-∞x f(u,v) dudv
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f(x,y) = ∂2/∂x∂y F(x,y)
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Marginal
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f(x) = ∑y∈Y f(x,y)
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f(x) = ∫y∈Y f(x,y)dy
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F(x) = lim y->∞ F(x,y) F(x) = ∫-∞x∫-∞∞ f(u,y) dydu
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E[g(x,y)] = ∑x,y∈XxY g(x,y)f(x,y)
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E[g(x,y)] = ∫x,y∈XxY g(x,y)f(x,y) dxdy
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Covariance
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indie => E[XY] = E[X]E[Y], Cov = 0 => ρ = 0
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ρ = 0 => E[XY] = E[X]E[Y]
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Cov[X,Y] = E[(X − E[X])(Y − E[Y])]
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Cov[X,Y] = E[XY] - E[X]E[Y]
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Correlation
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Gives linear relationship (+/-). |ρ| close to 1 is strong, close to 0 is weak
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special for bi-variate normal, indie <=> uncorrelated
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ρ(X,Y)= Cov[X,Y] / sqrt(Var[X]Var[Y])
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Conditional distribution
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The conditional distribution function of Y given X, written FY |x(·|x), is defined by
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F(y|x) = ∫-∞y f(x,v)/f(x) dv
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f(y|x) = f(x,y)/f(x) where f(x) = ∫-∞∞ f(x,y) dy
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Fy|x(y|x) = P(Y ≤ y|X = x)
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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
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f(y|x) = P(Y = y|X = x)
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f(x,y) = f(x|y)f(y) or f(y|x)f(x)
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Conditional expectation
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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
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E[h(X)g(Y)] = E[E[g(Y)|X]h(X)] = ∫(∫g(Y)f(Y|X) dx) h(X)f(x) dx
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E[Y|X =x] = ∑y∈Y yf(y|x)
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E[E[Y|X]] = E[Y]
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E[E[Y|X]g(X)] = E[Yg(X)]
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E[(aX + bY)|Z] = aE[X|Z] + bE[Y|Z]
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if X and Y are independent
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E[X|Y] = E[X]
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Var[X|Y] = E[X2|Y] - E[X|Y]2
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Bayes Theorem
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Consider probability space (Ω, F , P) and let A, B ∈ F with P(A), P(B) > 0. Then we have:
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P(B|A) = P(A|B)P(B) / P(A)
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Independence
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If X and Y are indie RV and g : X → R, h : Y → R, then the RV g(X) and h(Y ) are also indie
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Expectations
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1. if X≥0, E[X]≥0
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2. if a, b∈R then E[aX+bY]=aE[X]+bE[Y]
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3. if X = c∈R always, then E[X]=c.
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Variance
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1. For a ∈ R, Var[aX] = a2Var[X]
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2. Uncorrelated Var[X + Y] = Var[X] + Var[Y]
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Conditional Expectation
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Conditional expectations satisfies E[E[Y|X]] = E[Y] assuming all the expectations exist
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for any g : R → R, E[E[Y|X]g(X)] = E[Yg(X)] assuming all expectations exist
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Change of variable
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If (X1,X2) have joint density f(x,y) on Z, then for (Y1,Y2) = T(X1,X2), with T as described above, the joint density of (Y1,Y2), denoted g is: g(y1,y2)=f(T−1(y1,y2),T−1(y1,y2)) |J(y ,y )| (y1,y2)∈T
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