# Thread: p.d.f.s and c.d.f.s of functions

1. ## p.d.f.s and c.d.f.s of functions

Let X be a continuous random variable taking values in (a, b) with cumulative distribution function 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 from a uniform distribution on (0,1) to simulate a random sample from

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

Not reeally sure at all on this one..

2. Originally Posted by James0502
Let X be a continuous random variable taking values in (a, b) with cumulative distribution function 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 from a uniform distribution on (0,1) to simulate a random sample from

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

Not reeally sure at all on this one..

Use $F^{-1}(u)$ as your random sample, where $u$ is a computer-generated (pseudo-)random number drawn from a Uniform(0,1) distribution.

You need to find a formula for $F^{-1}(u)$ in order to make this practical.

(This is a general method for generating numbers from arbitrary distributions on a computer.)

3. how would I show the distribution is uniform though?

4. Originally Posted by James0502
Let X be a continuous random variable taking values in (a, b) with cumulative distribution function 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 from a uniform distribution on (0,1) to simulate a random sample from

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

Not reeally sure at all on this one..

5. "Let X be a continuous random variable taking values in (a, b) with cumulative distribution function F, strictly increasing
on (a, b). Show that Y = F(X) has a uniform distribution on (0, 1)."

The above statement is ture for all pdf's/cdf's, and it is the basis of simulation: generating data for numerous distributions.

Already mention but here how it works:
1. generate a random number in (0,1). (Use "=rand()" in MS Excel)
2. use F-inverse to compute the dist. value you want to generate.

I don't remember if Normal has an inverse (I think it should). But there is a way of generating Normal-ly distributed values buy above steps. I think they use "characteristic func", which unique for every pdf.

-O

6. Originally Posted by James0502
Let X be a continuous random variable taking values in (a, b) with cumulative distribution function F, strictly increasing
on (a, b). Show that Y = F(X) has a uniform distribution on (0, 1).

[snip]
Since F is strictly increasing, F has an inverse.

Let $0 \leq y \leq 1$. Then

$F(x) \leq y$ if and only if $x \leq
F^{-1}(y)$

so

$P(F(x) \leq y) = P(x \leq F^{-1}(y)) = F(F^{-1}(y)) = y$

I.e., the CDF of Y is Y for $0 \leq Y \leq 1$. So Y has a Uniform(0,1) distribution.