1. ## C.D.F

Let X be a continuous random variable taking values in (a, b) with c.d.f. F, strictly increasing on (a, b). Show that Y = F(X) has a uniform distribution on (0, 1). How would you use a set of computer generated random numbers (assumed to be drawn form a uniform distribution on
(0, 1) ), to simulate a random sample from

f(x) = (1/a)e^(-x/a) ,x>0?

I'm not quite sure how to do this, i'm fairly sure that the first bit is do with the fact that as it is strictly increasing there is a 1 to 1 mapping. I know that if i can show that P(Y<y)=y then i have demonstrated it but i'm not sure how?

I have no idea how to start the second bit. Any hints would be greatly appreciated.

Thanks.

2. Originally Posted by DeFacto
Let X be a continuous random variable taking values in (a, b) with c.d.f. F, strictly increasing on (a, b). Show that Y = F(X) has a uniform distribution on (0, 1). How would you use a set of computer generated random numbers (assumed to be drawn form a uniform distribution on
(0, 1) ), to simulate a random sample from

f(x) = (1/a)e^(-x/a) ,x>0?

I'm not quite sure how to do this, i'm fairly sure that the first bit is do with the fact that as it is strictly increasing there is a 1 to 1 mapping. I know that if i can show that P(Y<y)=y then i have demonstrated it but i'm not sure how?

I have no idea how to start the second bit. Any hints would be greatly appreciated.

Thanks.