Joint probability density function problem

  • #1
162
0
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
 

Answers and Replies

  • #2
2,112
18
For each permutation [itex]\sigma\in S_n[/itex] you can define a following subset of [itex]\mathbb{R}^n[/itex].

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

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

[tex]
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),
[/tex]

what information does the restriction of [tex]f|_{X_{\sigma}}[/tex] contain?

This is all related to your problem.
 
Last edited:

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