1. Combinations of random variables

Can anyone help me on the following Pls?

The random variable X has p.d.f given by
f(x) = x/2 0£ x £ 2
0 otherwise
(a) Find the c.d.f of X, FX(x).
The random variable Y = X2 and takes values over the range
0 £ y £ 4.
(b) Show that P(Y < y) = P(X £ Öy).
(c) Hence show that the c.d.f of Y, FY(y) is given by

FY(y) = 0, y < 0
y/4, 0 £ y £ 4
1, y > 4

2. Originally Posted by Sashikala
Can anyone help me on the following Pls?

The random variable X has p.d.f given by
f(x) = x/2 0£ x £ 2
0 otherwise
(a) Find the c.d.f of X, FX(x).
The random variable Y = X2 and takes values over the range
0 £ y £ 4.
(b) Show that P(Y < y) = P(X £ Öy).
(c) Hence show that the c.d.f of Y, FY(y) is given by

FY(y) = 0, y < 0
y/4, 0 £ y £ 4
1, y > 4
Set up and do the required integrations. What have you tried? Where do you get stuck?

3. for the part(a) I got
F(x)=0 , x< 0
square of x/4, 0£ x £ 2
1, x>2

Do we have to prove (b) with the
help of probability distribution table?
How to prove it using the p.d.f though
both are equal?
How to get at c.d.f of Y?

4. Originally Posted by Sashikala
for the part(a) I got
F(x)=0 , x< 0
square of x/4, 0£ x £ 2
1, x>2

Do we have to prove (b) with the
help of probability distribution table?
How to prove it using the p.d.f though
both are equal?
How to get at c.d.f of Y?
$\displaystyle F_Y(y) = \Pr(Y < y) = \Pr(X^2 < y) = \Pr(-\sqrt{y} < X < \sqrt{y})$ $\displaystyle = \Pr(X < \sqrt{y})$

since $\displaystyle \Pr(X < 0) = 0$

$\displaystyle = F_X(\sqrt{y}) = \frac{y}{4}$.

$\displaystyle f_Y(y) = \frac{dF_Y}{dy}$.

The details are left for you to fill in.

5. Combinations of random variables

Yes. I understood.
Thanks very much