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1 Joint Distributions Joint Probability Mass Function If X and Y are discrete random variables, the joint probability mass function (pmf) of X and Y is for all x, y ∈ RR p(x, y) = P (X = x, Y = y) Sometimes write pX,Y (x, y) for emphasis. Recall: P (X = x, Y = y) = P (X = x and Y = y) = P (X = x ∩ Y = y) = P ({ω ∈ Ω : X(ω) = x} ∩ {ω ∈ Ω : Y (ω) = y}). X X p(x, y) = P (X ∈ Range(X), Y ∈ Range(Y )) = 1. Note: x∈Range(X) y∈Range(Y ) If B ⊆ R2 is is the set of all pairs (x, y) that have a certain property, then X p(x, y) P ((X, Y ) ∈ B) = (x,y)∈B The pmfs of X and Y are obtained from the joint pmf of X and Y by X X pX (x) = pX,Y (x, y) and pY (x) = y∈Range(Y ) pX,Y (x, y) x∈Range(X) Proof: X pX (x) = P (X = x) = P (X = x, Y ∈ Range(Y )) = P (X = x, Y = y) = y∈Range(Y ) X pX,Y (x, y) y∈Range(Y ) pX and pY are sometimes called the marginal pmfs of X and Y , to distinguish them from the joint pmf pX,Y . Joint Probability Density Function If there exists a non-negative function f : R2 → R such that Z dZ b P (a ≤ X ≤ b, c ≤ Y ≤ d) = f (x, y)dxdy c a for all a, b, c, d ∈ R, then X and Y are called jointly continuous and f is called the joint probability density function (pdf) of X and Y . Sometimes write fX,Y (x, y) for emphasis. Z ∞Z ∞ Note: f (x, y)dxdy = P (X ∈ (−∞, ∞), Y ∈ (−∞, ∞)) = 1 −∞ −∞ If B ⊆ R2 is the set of all pairs (x, y) that have a certain property, then Z Z P ((X, Y ) ∈ B) = f (x, y)dxdy. B 2 If X and Y are jointly continuous with joint pdf fX,Y , then X and Y are continuous with pdfs Z ∞ Z ∞ fX,Y (x, y)dx fX,Y (x, y)dy and fY (y) = fX (x) = −∞ −∞ Proof: By definition, fX is what we integrate to get probabilities of X. We have Z b Z ∞ fX,Y (x, y)dy dx. P (a ≤ X ≤ b) = P (a ≤ X ≤ b, −∞ < Y < ∞) = a So fX (x) = R∞ −∞ fX,Y (x, y)dy −∞ is what we integrate to get probabilities of X. fX and fY are sometimes called the marginal pdfs of X and Y , to distinguish them from the joint pdf fX,Y . More Than Two Random Variables The joint pmf/pdf of X1 , . . . , Xn is defined similarly. For example, the joint pmf of discrete random variables X, Y, Z is pX,Y,Z (x, y, z) = P (X = x, Y = y, Z = z). If we want marginal distributions, pX,Y (x, y) = X pX,Y,Z (x, y, z), z∈Range(Z) pX (x) = X X y∈Range(Y ) z∈Range(Z) pX,Y,Z (x, y, z).