Question

(b) Let (X, Y) be a continuous vector random variable with joint probability density function denoted by f_{X,Y}(x, y) which is non-zero on the region given by {(x, y) : x > 0 and 0 < y < 1} and equals zero otherwise. (i) Find the joint probability density function of the vector (U, V) where U = X and V = XY. (ii) Use part (i) to show that V has probability density function given by f_{V}(v) = int_{v}^{infty} frac{1}{u} f_{X,Y} (u, frac{v}{u}) du, if v > 0, and f_{V}(v) = 0 otherwise. Give a simplification of this formula in the case when X and Y are independent. (iii) Let X and Y be independent continuous random variables where X has probability density function f_{X}(x) = egin{cases} frac{1}{2} x^{2} e^{-x} & ext{if } x > 0, \ 0 & ext{otherwise,} end{cases} and Y has probability density function f_{Y}(y) = egin{cases} 2y & ext{if } 0 < y < 1, \ 0 & ext{otherwise,} end{cases} Find the probability density function of V = XY and identify as a known distribution.

          (b) Let (X, Y) be a continuous vector random variable with joint probability density function denoted by f_{X,Y}(x, y) which is non-zero on the region given by {(x, y) : x > 0 and 0 < y < 1} and equals zero otherwise.
(i) Find the joint probability density function of the vector (U, V) where U = X and V = XY.
(ii) Use part (i) to show that V has probability density function given by
f_{V}(v) = int_{v}^{infty} frac{1}{u} f_{X,Y} (u, frac{v}{u}) du, if v > 0,
and f_{V}(v) = 0 otherwise. Give a simplification of this formula in the case when X and Y are independent.
(iii) Let X and Y be independent continuous random variables where X has probability density function
f_{X}(x) = egin{cases} frac{1}{2} x^{2} e^{-x} & 	ext{if } x > 0, \ 0 & 	ext{otherwise,} end{cases}
and Y has probability density function
f_{Y}(y) = egin{cases} 2y & 	ext{if } 0 < y < 1, \ 0 & 	ext{otherwise,} end{cases}
Find the probability density function of V = XY and identify as a known distribution.

$(b) Let (X, Y) be a continuous vector random variable with joint probability density function denoted by fX,Y(x, y) which is non-zero on the region given by (x, y) : x > 0 and 0 < y < 1 and equals zero otherwise. (i) Find the joint probability density function of the vector (U, V) where U = X and V = XY. (ii) Use part (i) to show that V has probability density function given by fV(v) = intv^infty frac1u fX,Y (u, fracvu) du, if v > 0, and fV(v) = 0 otherwise. Give a simplification of this formula in the case when X and Y are independent. (iii) Let X and Y be independent continuous random variables where X has probability density function fX(x) = egincases frac12 x^2 e^-x extif x > 0, 0 extotherwise, endcases and Y has probability density function fY(y) = egincases 2y extif 0 < y < 1, 0 extotherwise, endcases Find the probability density function of V = XY and identify as a known distribution.$

Added by Austin B.

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