Page 79 - Fundamentals of Probability and Statistics for Engineers

62 Fundamentals of Probability and Statistics for Engineers

which is expected. It gives the relationship between the joint jpmf and the
conditional mass function. As we will see in Example 3.9, it is sometimes more
convenient to derive joint mass functions by using Equation (3.35), as condi-
tional mass functions are more readily available.
If random variables X and Y are independent, then the definition of inde-
pendence, Equation (2.16), implies

p XY xjy†ˆ p x†; 3:36†
X
and Equation (3.35) becomes

p XY x; y†ˆ p x†p y†: 3:37†
X
Y
Thus, when, and only when, random variables X and Y are independent, their
jpmf is the product of the marginal mass functions.
Let X be a continuous random variable. A consistent definition of the
conditional density function of X given Y ˆ y, f 9xjy), is the derivative of
XY
its corresponding conditional distribution function. Hence,

dF XY xjy†
f XY xjy†ˆ ; 3:38†
dx

j
where F XY (x y) is defined in Equation (3.33). To see what this definition leads
to, let us consider
P x 1 < X x 2 \ y 1 < Y y 2 †
P x 1 < X x 2 jy 1 < Y y 2 †ˆ : 3:39†
P y 1 < Y y 2 †

In terms of jpdf f (x, y), it is given by
XY
y 2 x 2 y 2
Z Z Z Z 1
P x 1 < X x 2 jy 1 < Y y 2 †ˆ f x;y†dxdy f x;y†dxdy
XY XY
1
y 1 x 1 y 1
Z Z Z
y 2 x 2 y 2
ˆ f x;y†dxdy f y†dy: 3:40†
XY Y
y 1 x 1 y 1
By setting x 1 ˆ 1, x 2 ˆ x, y 1 ˆ y,a nd y 2 ˆ y ‡ y, and by taking the limit
y ! 0, Equation (3.40) reduces to
Z x
f XY u; y† du
F XY xjy†ˆ 1 ; 3:41†
f y†
Y
6ˆ
provided that f (y) 0.
Y

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