Transformations of Random Variables - Advanced Probability Theory for Biomedical Engineers - page 70

ψ

1

γ

0 1 2

FIGURE 6.10: Support region for Example 6.4.6.

Example 6.4.6. Random variables x and y have the joint PDF

4

αβ, 0 <α< 1 , 0 <β< 1

f x , y ( α, β ) =

0 , otherwise .

Find the PDF for z = g ( x , y ) = x + y .

Solution. Let auxiliary variable w = h ( x , y ) = y . Solving γ = α + β and ψ = β , we find

α = γ − ψ and β = ψ .Wefind

=

11
01

= 1 ,

J

so that

4(

γ − ψ ) ψ, 0 <γ − ψ< 1 , 0 <ψ< 1

f z , w ( γ,ψ ) = f x , y ( γ − ψ, ψ ) =

0 , otherwise .

The support region for f z , w is shown in Figure 6.10. Referring to Figure 6.10, for 0 <γ < 1,

γ

2
3 γ

3

f z (

γ ) = 4( γ − ψ ) ψ ψ = .

d

0

For 1 <γ < 2,

1

2
3 γ

8
3 .

3

f z ( γ ) = 4( γ − ψ ) ψ d ψ =− + 4 γ −

γ − 1

Otherwise, f z = 0.

Conditional PDF Technique
Since a conditional PDF is also a PDF, the above techniques apply to find the conditional PDF
for z

=

g ( x

,

y ), given event A . Consider the problem where random variables x and y have joint