Question

The joint pdf of two dimensional random variables (X, Y) is given by f(x,y) = kxye-(x+y); ;x>0, y > 0. Find the value of k and prove that X and Y are independent.

Answer 1

Answer:

Let XX and YY be jointly continuous random variables with joint PDF

fX,Y(x,y)=⎧⎩⎨⎪⎪cx+10x,y≥0,x+y<1otherwisefX,Y(x,y)={cx+1x,y≥0,x+y<10otherwise

Show the range of (X,Y)(X,Y), RXYRXY, in the x−yx−y plane.

Find the constant cc.

Find the marginal PDFs fX(x)fX(x) and fY(y)fY(y).

Find P(Y<2X2)P(Y<2X2).