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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).