# Joint Density

Discussion in 'Probability' started by Tims, Oct 2, 2010.

1. ### TimsGuest

Let X1,X2,X3, and X4 be independent continuous random variables with
a common distribution function F and let
p = P{X1 <X2 >X3 <X4}

(a)
Find p by integrating the joint density function over the appropriate
region.

(b)
Find p by using the fact that all 4! possible orderings of X1, . . . ,
X4 are equally likely.

[Difficulty]
I have starting problem with (a)

(b) seems to be easy

[Thoughts]
For (b),

X1 __ X2 __ X3 __ X4

The symbols can be arranged in 4! ways.
The signs in each blank can be either <, = or >=

The blanks need to be filled with <, > and < respectively,
SO, P(<) = P(>) = P(=) = 1/3

So (b) must be (1/3)^3 times 4!

Tims, Oct 2, 2010