Page 220 - Probability and Statistical Inference

4. Functions of Random Variables and Sampling Distribution 197

Let Y = X and Y = X + X . We first wish to obtain the joint pdf of Y and Y .
2
1
1
1
2
1
2
Then, the goal is to derive the marginal pdf’s of Y , Y .
1 2
The one-to-one transformation (x , x ) → (y , y ) leads to the inverse: x =
2
1
2
1
1
y , x = y – y so that |det(J)| = 1. Now, x > 0 implies that 0 < 2y < y < ∞
1
2
1
2
1
2
2
since y < y – y . Thus, (4.4.4) leads to the following joint pdf of Y and Y :
1 2 1 1 2
The marginal pdf’s of Y , Y can be easily verified as the following:
1 2
We leave out some of the intermediate steps as the Exercise 4.4.6.
Example 4.4.8 Suppose that X and X have their joint pdf given by
1 2
Let Y = X + X and Y = X – X . We first wish to obtain the joint pdf of Y 1
2
2
1
2
1
1
and Y . Then, the goal is to derive the marginal pdf’s of Y , Y .
2 1 2
The one-to-one transformation (x , x ) → (y , y ) leads to the inverse: x =
1
1
1
2
2
1/2(y + y ), x = 1/2(y – y ) so that |det(J)| = 1/2. Observe that: 0 < x < 1 ⇒
1
2
2
1
1
2
0 < y + y < 2; 0 < x < 1 ⇒ 0 < y – y < 2; 0 < x + x < 1 ⇒ 0 < y < 1. Let
2
1
1
2
2
1
2
1
γ = {(y , y ) ∈ ℜ : 0 < y < 1, 0 < y + y < 2, 0 < y – y < 2}. The joint pdf of
2
1
1
2
1
2
1
Y and Y is then given by
1 2
The marginal pdf’s of Y , Y can be easily verified as the following:
1 2
We leave out some of the intermediate steps as the Exercise 4.4.7. !
Example 4.4.9 The Helmert Transformation: This consists of a very
special kind of orthogonal transformation from a set of n iid N(µ, σ )
2