# The Joint PDF of Two Uniform Distributions

1. Oct 16, 2015

### tamuag

1. The problem statement, all variables and given/known data
A manufacturer has designed a process to produce pipes that are 10 feet long. The distribution of the pipe length, however, is actually Uniform on the interval 10 feet to 10.57 feet. Assume that the lengths of individual pipes produced by the process are independent. Let X and Y represent the lengths of two different pipes produced by the process.

What is the joint pdf for X and Y?

2. Relevant equations

3. The attempt at a solution
My roommate says the answer is $f(x,y) = 1 / 0.57^2, 10 \leq x \leq 10.57, 10 \leq y \leq10.57$

I understand why $10 \leq x \leq 10.57, 10 \leq y \leq 10.57$, but why is $f(x,y) = 1 / 0.57^2$?

My thoughts so far:
Isn't $f(x,y) = 1 / 0.57^2$ really just $f(x,y) = (1 / 0.57)(1 / 0.57)$, where $f(x) = f(y) = 1 / 0.57$?

What I really don't understand is why $f(x) = 1 / 0.57$ in the first place? Where does that term come from? I get that it's a constant because the distribution is uniform, but why is it a fraction and why is the denominator $0.57$?

2. Oct 16, 2015

### RUber

$f(x) = \frac{1}{0.57}$ because $\int_{10}^{10.57} f(x) dx = 1$ and $f(x)$ is constant.

3. Oct 16, 2015

### Ray Vickson

The random variables are uniform over intervals of length 0.57.