The Joint PDF of Two Uniform Distributions

[tamuag]

Homework Statement

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?

Homework Equations

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?

 

[RUber]

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

 

[Ray Vickson]

Homework Statement

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?

Homework Equations

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?

The random variables are uniform over intervals of length 0.57.