Transformations of Random Variables - Advanced Probability Theory for Biomedical Engineers - page 69

ψ

3

2

1

− 3 − 2 − 1

γ

1 2 3

− 1

− 2

− 3

FIGURE 6.9: Support region for Example 6.4.4.

Example 6.4.5. Random variables x and y have the joint PDF

4

αβ, 0 <α< 1 , 0 <β< 1

f x , y ( α, β ) =

0 , otherwise .

x 2 Find the PDF for z .

Solution. Let auxiliary variable w

= g ( x , y ) =

2

= h ( x , y ) = y . Solving γ = α and ψ = β , we find α =

± √ γ α = √ γ

and β = ψ . The only solution inside the support region for f x , y is and β = ψ .

The Jacobian of the transformation is

=

2 0

01

α

J = 2 α,

so that

4 √ γψ
2 √ γ

⎧
⎨
⎩

f x , y ( √ γ,ψ ) = 2 ψ, 0 <γ < 1 , 0 <ψ< 1

f z , w ( γ,ψ ) = =

2 √ γ

0 , otherwise .

We find the marginal PDF for z as

∞

1

f z ( γ ) = f z , w ( γ,ψ ) d ψ = 2 ψ d ψ.

−∞ 0

for 0 <γ < 1 and f z ( γ ) = 0, otherwise.