# Joint probability density function problem

1. Jan 14, 2009

Suppose that h is the probability density function of a continuous random variable.
Let the joint probability density function of X, Y, and Z be
f(x,y,z) = h(x)h(y)h(z) , x,y,zER

Prove that P(X<Y<Z)=1/6

I don't know how to do this at all. This is suppose to be review since this is a continuation course, but I didn't take the previous course

Any help would be greatly appreciated

2. Jan 14, 2009

### jostpuur

For each permutation $\sigma\in S_n$ you can define a following subset of $\mathbb{R}^n$.

$$X_{\sigma} := \{x\in\mathbb{R}^n\;|\; x_{\sigma(1)} \leq x_{\sigma(2)} \leq \cdots \leq x_{\sigma(n)}\}$$

Some relevant questions: Are these subsets (with different $\sigma$) mostly/essentially disjoint? In what sense they are the same shape? How many of these subsets are there? What is the union $$\bigcup_{\sigma\in S_n} X_{\sigma}$$? If a function $f:\mathbb{R}^n\to\mathbb{R}$ is invariant under a swapping of parameters like this

$$f(x_1,\ldots, x_n) = f(x_1,\ldots, x_{i-1},x_j, x_{i+1},\ldots x_{j-1}, x_i, x_{j+1},\ldots, x_n),$$

what information does the restriction of $$f|_{X_{\sigma}}$$ contain?

This is all related to your problem.

Last edited: Jan 14, 2009