Fundamentals of Probability and Statistics for Engineers

density function (jpdf) of two ra ndom variables, X and Y, is defined by the
partial derivative

Since FX Y (x,y) is monotone nondecreasing in both x and y,fX Y (x, y) is
nonnegative for all x and y. We also see from Equation (3.24) that

Moreover, with x 1 <x 2 ,andy 1 <y 2 ,

The jpdf fX Y (x, y) defines a surface over the (x, y) pl ane. As indicated by
Equation (3.26), the probability that random variables X and Y fall within a
certain region R is equal to the volu me under the surface of fX Y (x, y) and
bounded by that region. This isillustrated in F igure 3.13.

fXY(x, y)

x

y

R

Figure 3. 13 A joint probability density function, fX Y (x,y)

56 Fundamentals of Probability and Statistics for Engineers

fXY…x;y†ˆ

q^2 FXY…x;y†

qxqy

: … 3 : 24 †

FXY…x;y†P…Xx\Yy†ˆ

Zy

1

Zx

1

fXY…u;v†dudv: … 3 : 25 †

P…x 1 <Xx 2 \y 1 <Yy 2 †ˆ

Zy 2

y 1

Z x 2

x 1

fXY…x;y†dxdy: … 3 : 26 †