Function of Random Variables, and Function of Random Vectors
By: Adam Edelweiss
Tags: Random Variable, Probability, Mathematics, Statistics, Random Vector

Recall: Probability Functions

(1) Probability Mass Function (PMF)

(2) Probability Density Function (PDF)

(3) Cumulative Density Function (CDF)

(4) Joint Probability Density Function

(5) Marginal Probability Density Function

(6) Joint Cumulative Density Function

(7) Joint Probability Mass Function

(8) Marginal Probability Mass Function

(9) Definition of Independent Random Variables

(10) Properties of Independency

2. Function of Random Variables

(1) CDF Yielding Multiplied Continuous Random Variables

If W = aX, then,

(2) PDF Yielding Multiplied Continuous Random Variables

If W = aX, then,

(3) CDF Yielding Continuous Random Variables Adding Constant

If W = X+b, then,

(4) PDF Yielding Continuous Random Variables Adding Constant

If W = X+b, then,

(5) CDF of the Sum of Two Random Variables

If W = X+Y, then,

(6) PDF of the Sum of Two Random Variables

If W = X+Y, then,

(7) PDF of a Function with Two Random Variables

If W = g(X, Y), then,

(8) PDF of the maximum of Two Random Variables

If W = max{X, Y}, then,

(9) CDF of an Inverse Uniform Random Variable

If W = F^(-1)(X) and X ~Unif(0, 1), then,

3. PMF and PDF with Conditional Event

(1) Conditional Joint PMF

For any given event B, a region of X, Y plane with ℙ(B) > 0,

Or 0 otherwise.

(2) Conditional Joint PDF

For any given event B, given a joint function f(x, y) with ℙ(B) > 0,

Or 0 otherwise.

(3) Conditional Expected for Discrete Random Variables

If W = g(X, Y), then,

(4) Conditional Expected for Continuous Random Variables

If W = g(X, Y), then,

4. PMF and PDF with Conditional Random Variable

(1) Conditional PMF

For any event Y = y such that p(y) > 0, then the conditional PMF of X given Y = y is,

This is also,

(2) Conditional PDF

For y such that f(y) > 0, then the conditional PDF for X given {Y = y} is,

(3) Conditional Expected for Discrete Random Variables

If W = g(X, Y), then,

(4) Conditional Expected for Continuous Random Variables

If W = g(X, Y), then,

5. Random Vectors

(1) The Definition of the Random Vector

Suppose X1, X2, …, Xn are random variables, then the vector X = [X1 X2 … Xn]’ is then defined as a random vector. When X1 = x1, X2 = x2, …, Xn = xn, then the vector x = [x1 x2 … xn]’ is then defined as a sample of the random vector.

(2) PMF for a Random Vector

The PMF for a random vector X is,

(3) PDF for a Random Vector

The PDF for a random vector X is,

(4) CDF for a Random Vector

The CDF for a random vector X is,

(5) PMF of a Function of Discrete Random Vector

For random variable W = g(X),

(6) PDF of a Function of Discrete Random Vector

For random variable W = g(X),

(7) Expectation of a Function of a Discrete Random Vector

For random variable W = g(X),

(8) Expectation of a Function of a Continuous Random Vector

For random variable W = g(X),

(9) PDF of a Function Yielding Random Vector

For random vector Y = AX + b, and X is a continuous random vector and A is an invertible matrix, then the PDF of Y is,

(10) Expectation of Random Vector

The expectation value of a random vector X is a column vector, then,

(11) Correlation Matrix of Random Vector

The correlation matrix of a random vector X is an n × n matrix R, which is defined as,

(12) Variance-Covariance (Cross-Correlation) Matrix of Random Vector

The variance-covariance matrix of a random vector X is an n × n matrix C, which is defined as,