Let X be a continuous random variable with PDF fx Let Y gX

Let X be a continuous random variable with PDF: f(x) = Let Y = g(X) = -X^2. (a) What are Rx and Ry the support of X and Y)? (b) Is g(X) monotonic? (c) Find fy(y).

Solution

(a)

The support of a continuous random variable X is {x ? R : f(x) is not equal to 0}.

Hence , R_x (support of X) = [0,2]

and R_y (support of Y) = (0,2]

(b)

In the interval (0,2) , g(x) is monotoniacally decreasing function.

(c)

If g is monotone decreasing, then

F_Y(y) = P(Y ? y) = P(g(X) ? y) = P(X ? g