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5-1 TWO DISCRETE RANDOM VARIABLES 145

Definition
If X and Y are discrete random variables with joint probability mass function f (x, y),
XY
then the marginal probability mass functions of X and Y are
1x2 P1X x2 a f 1x, y2 and f 1 y2 P1Y y2 a f 1x, y2
f X XY Y XY
R x R y
(5-2)
where R denotes the set of all points in the range of (X, Y) for which X x and
x
R denotes the set of all points in the range of (X, Y) for which Y y
y

Given a joint probability mass function for random variables X and Y, E(X) and V(X) can
be obtained directly from the joint probability distribution of X and Y or by first calculating the
marginal probability distribution of X and then determining E(X ) and V(X ) by the usual
method. This is shown in the following equation.

Mean and
Variance from If the marginal probability distribution of X has the probability mass function f (x),
X
Joint then
Distribution
E1X 2 a x f X 1x2 a x a a f XY 1x, y2b a a x f XY 1x, y2
X
x x R x x R x
a x f 1x, y2 (5-3)
XY
R
and

2
2
2
V1X 2 X a 1x 2 f X 1x2 a 1x 2 a f XY 1x, y2
X
X
x x R x
2
2
a a 1x 2 f XY 1x, y2 a 1x 2 f XY 1x, y2
X
X
x R x R
where R x denotes the set of all points in the range of (X, Y) for which X x and R
denotes the set of all points in the range of (X, Y)
EXAMPLE 5-4 In Example 5-1, E(X) can be found as
E1X 2 03 f XY 10, 02 f XY 10, 12 f XY 10, 22 f XY 10, 32 f XY 10, 424
13 f XY 11, 02 f XY 11, 12 f XY 11, 22 f XY 11, 324
12, 02 f 12, 12 f 12, 224
23 f XY XY XY
33 f XY 13, 02 f XY 13, 124
43 f XY 14, 024
030.00014 130.00364 230.04864 330.029164 430.65614 3.6

Alternatively, because the marginal probability distribution of X is binomial,

E1X 2 np 410.92 3.6