# Probability of a given ordering of independent random variables

I ran into the following problem and I can't figure out a way of solving it... Suppose I have $N$ continuous random variable $X_1$,...,$X_N$ that are independent, but their law is different, and I know the cdf of each of them, i.e. the cdf of $X_i$ is $F_i(x)$.

I want to compute the probability of a given ordering of those random variable, for example

$$\mathcal{P}\left[X_1<X_2<...<X_N\right].$$

Is there some simplified approach to do this, something else than the brute force multidimensional integral ? For $N=2$ it is easy to see that it is worth $\mathbb{E}[F_1(X_2)]$, but I wonder if there is some kind of generalization for $N>2$.

Thank you

Kapouet, since my comments are long and exceed the number of allowed characters in a regular comment, i had to put it so (i.e., "an answer"). anyway, anyone can edit the comments and even delete them. Here are some thoughts:

(1) suppose there were a genealization with a simple form, what would you expect it to be?

(2) say, you have to do the brutal force intergation, then any systematic reduction of the integral will necessarily require some invariant property of the joint distribution of the random variables. recall the permutation invariance of the joint distribution of indepenent and identically distributed random variables.

(3) since the random variables are independent and the set is induced by succecive "cuts" (compare it with the case $N=2$), you can try mathematical induction to get a general formula without possibly computing the multiple intergal by brutal force. specifically, try a conditional argument on one or more of the random variables and combine induction. this way you are very likely to get some elegant and general formula

(4) finally, define the left inverse from $F_i$ and convert the problem to the case of random variables on the unit cube (if possible, say, when $F_i$ are absolutely continuous). then you proceed with (3)

Good luck!

Comments (4/22/2017): in my comments, i did not assume that the random variables are identically distributed; i was just referring to the i.i.d. case to illustrate that some invariant properties of the joint distribution or the set for which the probability is computed is needed for a systematic reduction of the integral.

This type of thing is usually hard to compute and one usually does not have an answer for this at hand unless he/she has done research along this line. i am confident that the strategy posted is the correct way to go! however, it is unlikely that someone else will compute the probability for the question owner since we are busy and sometimes very busy ....

• The question says the laws are different. You can't assume they are identically distributed. – Douglas Zare Apr 22 '17 at 7:03
• hi Douglas, in my comments, i did not say the random variables are identically distributed. instead i was referring to the analogy for the identically distributed case to illustrate that a systematic reduction usually requires some invariant property of the joint distribution or of the set for which the probability is computed – Chee Apr 22 '17 at 12:02
• Hi Chee, thank you for your thoughts, it's much appreciated! – kapouet Apr 22 '17 at 14:33
• Hi Kapouet, it is very likely that you will get a recursion formulae for the probability with respect to N – Chee Apr 22 '17 at 14:48