Discrete Multivariate R.V.s
Definition (Joint CDF) The Joint CDf of r.v.s X and Y is the function FX,Y given by
FX,Y(x,y)=P(X≤x,Y≤y)Definition (Joint PMF) The Joint PMF of discrete r.v.s X and Y is the function pX,Y given by
pX,Y(x,y)=P(X=x,Y=y)Definition (Marginal PMF) For discrete r.v.s X and Y, Marginal PMF of X is
P(X=x)=∑yP(X=x,Y=y)Definition (Conditional PMF) For discrete r.v.s X and Y, the Conditional PMF of X given Y=y is
PX|Y(x|y)=P(X=x|Y=y)=P(X=x,Y=y)P(Y=y)Definition (Independence of Discrete R.V.s) Random variables X and Y are independent if for all x and y
FX,Y(x,y)=FX(x)FY(y)for all x and y also equivalent to the condition
P(Y=y|X=x)=P(Y=y)
Continuous Multivariate R.V.s
Definition (Joint PDF) If X and Y are continuous with joint CDF FX,Y then
fX,Y(x,y)=∂2∂x∂yFX,Y(x,y)Definition (Marginal PDF) If X and Y are continuous with joint PDF fX,Y then
fX(x)=∫∞−∞fX,Y(x,y)dyDefinition (Conditional PDF) For continuous r.v.s. X and Y with joint PDF fX,Y the Conditional PDF of Y given X=x is
fY|X(y|x)=fX,Y(x,y)fX(x)Definition (Independence of Continuous R.V.s) Random variables X and Y are independent if for all x and y
FX,Y(x,y)=FX(x)FY(y)If X and Y are continuous with joint PDF fX,Y
fX,Y(x,y)=fX(x)fY(y)Theorem (2D LOTUS) Let g be a function from R2 to R
If X and Y are discrete
E(g(X,Y))=∑x∑yg(x,y)P(X=x,Y=y)If X and Y are continuous
E(g(X,Y))=∫∞−∞∫∞−∞g(x,y)fX,Y(x,y)dxdyGeneral Bayes’ Rule

Convariance and Correlation
Covariance
- Measure a tendency of two r.v.s X&Y to go up or down together
- Positive Covariance: X go up, Y tends go up
- Negative Covariance: X go up, Y tends go down
Definition (Covariance) The covariance between r.v.s X and Y is
Cov(X,Y)=E((X−EX)(Y−EY))=E(XY)−E(X)E(Y)Theorem (Uncorrelated) If X and Y are independent, then they are Uncorrelated(Cov(X,Y)=0)
Properties of Covariance
- Cov(X,X)=Var(X)
- Cov(X,Y)=Cov(Y,X)
- Cov(X,c)=0
- Cov(a⋅X,Y)=a⋅Cov(X,Y)
- Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Z)
- Cov(X+Y,W+Z)=Cov(X,Z)+Cov(X,W)+Cov(Y,Z)+Cov(Y,W)
- Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)
- For n r.v.s X1,⋯,Xn Var(X1+⋯+Xn)=Var(Xa)+⋯+Var(Xn)+2∑i<jCov(Xi,Yj)
Definition (Correlation) The Correlation between r.v.s X and Y is
Corr(X,Y)=Cov(X,Y)√Var(X)Var(Y)Shifting and Scaling X and Y has no effect on correlation

Theorem (Correlation Bounds) For any r.v.s X and Y
−1≤Corr(X,Y)≤1
Change of Variables
Theorem (Change of Variables in One Dimension) Let X be a continuous r.v. with PDF fX, and let Y=g(X), where g is differentiable and strictly increasing. Then the PDF of Y is given by
fY(y)=fX(x)∣∣∣dxdy∣∣∣where x=g−1(y)
Proof:
FY(y)=P(Y≤y)=P(g(X)≤y)=P(X≤g−1(y))=FX(g−1(y))=FX(x)Then result obtained By the chain rule
Theorem (Change of Variables) Let X=(X1,…,Xn) be a continuous random vector with joint PDF fX(x) and Y=g(X), g is an invertible function from Rn to Rn then ∂x∂y form a Jacobian matrix
∂x∂y=⎛⎜
⎜
⎜
⎜⎝∂x1∂y1∂x1∂y2⋯∂x1∂yn⋮⋮⋮∂xn∂y1∂xn∂y2⋯∂xn∂yn⎞⎟
⎟
⎟
⎟⎠Then the joint PDF of Y is
fY(y)=fX(x)∣∣∣∂x∂y∣∣∣
Convolutions
Theorem (Convolution Sums and Integrals)
If X and Y are independent discrete r.v.s, then the PMF of their sum T=X+Y is
P(T=t)=∑xP(Y=t−x)P(X=x)=∑yP(X=t−y)P(Y=y)If X and Y are independent continuous r.v.s, then the PMF of their sum T=X+Y is
fT(t)=∫∞−∞fY(t−x)fX(x)dx=∫∞−∞fX(t−y)fY(y)dy
Order Statistics
Definition (Order Statistics) For r.v.s X1,X2,…,Xn the order statistics sre the random variables X(1),…,X(2), where
- X(1)=min(X1,…,Xn)
- X(2) is the 2nd of X1,…,Xn
- ⋮
- Xn=max(X1,…,Xn)
The order statistics are dependent, for example , if X(1)=100, then X(n) is forced to be ≥100
We foucs on the case X1,…,Xn are i.i.d continuous r.v.s, with CDF F and PDF f
Theorem (CDF of Order Statistics) Let X1,…,Xn be i.i.d continuous r.v.s with CDF F, Then the CDF of the jth order statistic X(j) is
P(X(j)≤x)=n∑k=j(nk)F(x)k(1−F(x))n−kProof:
Let’s start with a specical case when j=n,X(n)=max(X1,…,Xn):
FX(n)(x)=P[max(X1,...,Xn)≤x]=P(X1≤x)⋯P(Xn≤x)=[F(x)]n
Then, consider another special case when j=1,X(1)=min(X1,…,Xn):
FX(1)(x)=P[min(X1,...,Xn)≤x]=1−P(X1>x)⋯P(Xn>x)=1−[1−F(x)]nThe result here can be rewrite as ∑nk=1(nk)F(x)k(1−F(x))n−k
This result can be obtained by expand [F(x)+1−F(x)]n
Finally, let’s consider more general case where 1<j<n,X(j)≤x, this means at least j of {Xi} fall to the left of x
Denote N as the nunber of Xi landing to the left of x. Xi lands to the left of x w.p. P(Xi≤x)=F(x). Then N∼Bin(n,F(x))
P(X(j)≤x)=P(N≥j=n∑k=j(nk)F(x)k(1−F(x))n−k
Theorem (PDF of Order Statistic) Let X1,…,Xn be i.i.d. continuous r.v.s with CDF F and PDF f. Then the marginal PDF of jth order statistic X(j) is
fX(j)(x)=n(n−1j−1)f(x)F(x)j−1(1−F(x))n−jTheorem (Joint PDF) Let X1,…,Xn be i.i.d. continuous r.v.s with PDF f, Then the joint PDF of all order statistics is
fX(1),...,X(n)(x1,...,xn)=n!n∏i=1f(xi),x1<x2<⋯<xnExample 1(Order Statistics of Uniforms) U1,U2,…,Un are i.i.d. Unif(0,1) r.v.s with CDF F and PDF f
For 0≤x≤1 ,f(x)=1 , F(x)=x, Then
fU(j)=n(n−1j−1)xj−1(1−x)n−jFU(j)(x)=n∑k=j(nk)xk(1−x)n−k=∫x0fU(j)(t)dt=n!(j−1)!(n−j)!∫x0tj−1(1−t)n−jdt