Question

Other Problems 1. Let X1​ and X2​ be iid with pdf fXi​​(xi​)=1/xi2​ for 1≤xi​<[infinity],i=1,2. (a) Find the joint pdf of Y1​=X1​X2​ and Y2​=X1​. (b) Find the marginal pdf of Y1​.

Answer 1

To find the joint pdf of Y1 = X1 X2 and Y2 = X1, we can use the transformation method. The joint pdf of Y1 and Y2 is 1/Y2^4. To find the marginal pdf of Y1, we integrate out Y2 and get -1/3Y2^3.

Step-by-step explanation:

Joint PDF of Y1 and Y2

To find the joint pdf of Y1 = X1 X2 and Y2 = X1, we need to find their respective distributions first. Since X1 and X2 are independent and have pdf fXi(xi) = 1/xi^2 for 1≤xi<[infinity], we can use the transformation method to find the joint pdf of Y1 and Y2:

1. Find the transformations from (X1, X2) to (Y1, Y2): Y1 = X1 X2, Y2 = X1
2. Find the inverse transformations from (Y1, Y2) to (X1, X2): X1 = Y2, X2 = Y1/Y2
3. Find the Jacobian of the inverse transformations: J = |∂(X1, X2)/∂(Y1, Y2)| = |1/Y2 - Y1/Y2^2| = |1/Y2^2|
4. Use the Jacobian to find the joint pdf of Y1 and Y2: fY1,Y2(y1, y2) = fX1,X2(x1, x2) * |J| = 1/(Y2^2 * Y2^2) = 1/Y2^4

Marginal PDF of Y1

To find the marginal pdf of Y1, we integrate out Y2 from the joint pdf of Y1 and Y2:

1. Integrate fY1,Y2(y1, y2) with respect to Y2: ∫(1/Y2^4) dY2 = -1/3Y2^3
2. Because Y2 can take any positive value, the marginal pdf of Y1 is fY1(y1) = -1/3Y2^3