Question

Let X 1 ∽χ 2 (m,δ) and X 2 ∽χ 2 (n) where X 1

and X 2 are independently distributed. (a) Derive the joint probability density function (pdf) of Y 1
and Y 2 where X 1 =Y 1 Y2 and X 2 =Y 2 (1−Y 1 ) (b) Derive the marginal pdf of Y 1 in 3(a).

Answer 1

(a) The joint probability density function (pdf) of Y1 and Y2 is given by:

f_Y1Y2(y1, y2) = f_X1((y1 / (1 - y1))X2) * (X2 / (1 - y1)^2) * f_X2(y2)

(b) The marginal pdf of Y1 is obtained by integrating the joint pdf over Y2:

f_Y1(y1) = ∫ f_Y1Y2(y1, y2) dy2

Note: The specific form of the distributions f_X1 and f_X2 needs to be known to compute the marginal pdf of Y1.