Question

dem inrinkuut tant follen an expon ential dis hiti teen net ?

Answer 1

a. The joint pdf of (X, Y) is f(x,y)= f(x)f(y) = e-r-y, x.J > 0: since X and Y are independent f(x, y)d dy Now, r, y>00 0 or y 0 0 u, v The joint pdf of (U,V) is f(u, v) = ve-r: 0 < u < 1,v>0 0 otherwise c. The marginal pdf of U is fu(u) f(u, u)du = ve-udu-「(2) = 1: 0 < u < 1 0 0 otherwise Hence UUniform(0,1) d. The marginal pdf of V is 0 otherwise V ~ Gamma distribution(2)

$e.~The~conditional~distribution~of~U~given~v~is\\ ~f(u|v)=\frac{f(u,v)}{f_V(u)}=1~if~0<u<1\\ =0~otherwise\\ f.~The~conditional~distribution~of~V~given~u~is\\ ~f(v|u)=\frac{f(u,v)}{f_V(u)}=ve^{-v}~if~v>0\\ =0~otherwise\\$