Applied Statistics and Probability for Engineers

5-1 TWO DISCRETE RANDOM VARIABLES 145

Given a joint probability mass function for random variables Xand Y, E(X) and V(X) can

be obtained directly from the joint probability distribution of Xand Yor by first calculating the

marginal probability distribution of Xand then determining E(X) and V(X) by the usual

method. This is shown in the following equation.

If the marginal probability distribution of Xhas the probability mass function fX(x),

then

(5-3)

and

where Rxdenotes the set of all points in the range of (X, Y) for which Xxand R

denotes the set of all points in the range of (X, Y)

 a

x

a

Rx

1 xX 22 fXY 1 x, y 2  a

R

1 xX 22 fXY 1 x, y 2

V 1 X 2 ^2 X a

x

1 xX 22 fX 1 x 2  a

x

1 xX (^22) a
Rx
fXY 1 x, y 2
 a
R
x fXY 1 x, y 2
E 1 X 2 X a
x
x fX 1 x 2 a
x
x aa
Rx
fXY 1 x, y2b a
x
a
Rx
x fXY 1 x, y 2
Mean and
Variance from
Joint
Distribution
EXAMPLE 5-4 In Example 5-1, E(X) can be found as
Alternatively, because the marginal probability distribution of Xis binomial,
E 1 X 2 np 41 0.9 2 3.6
 03 0.0001 4  13 0.0036 4  23 0.0486 4  33 0.02916 4  43 0.6561 4 3.6
 43 fXY 1 4, 0 24
 33 fXY 1 3, 0 2 fXY 1 3, 1 24
 23 fXY 1 2, 0 2 fXY 1 2, 1 2 fXY 1 2, 2 24
 13 fXY 1 1, 0 2 fXY 1 1, 1 2 fXY 1 1, 2 2 fXY 1 1, 3 24
E 1 X 2  03 fXY 1 0, 0 2 fXY 1 0, 1 2 fXY 1 0, 2 2 fXY 1 0, 3 2 fXY 1 0, 4 24
If Xand Yare discrete random variables with joint probability mass function fXY(x,y),
then the marginal probability mass functionsof Xand Yare
(5-2)
where Rxdenotes the set of all points in the range of (X, Y) for which Xxand
Rydenotes the set of all points in the range of (X, Y) for which Yy
fX 1 x 2 P 1 Xx 2  a
Rx

fXY 1 x, y 2 and fY 1 y 2 P 1 Yy 2  a

Ry

fXY 1 x, y 2

Definition

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