Maximum Value of Multiple Uniformly Distributed Variables In a computer program I am writing, I have several independent variables, each of which is uniformly distributed between two real values. I seek the probability of each variable being the maximum of the group.
That is, let $X_1, X_2, X_3,\ldots, X_n$ be my random variables. I seek a probability function $P$ such that $P(X_m)$ is the probability that $X_m = \max\{X_1, X_2, X_3,\ldots, X_n\}$. How would I find such a function given the ranges of each random variable? I don't need the actual maximum value - rather, I only need the probabilities that each variable produces that maximum value.
Since this is for a computer program, I would like to have a solution which can be easily algebraically calculated for an arbitrary number of variables, each with arbitrary ranges. However, if this is not feasible, I would naturally accept a calculus based program.
 A: As I understand it, the interval of each $X_i$ is a parameter of your algorithm.
Let $X_1,X_2...,X_n$ be the random variables that survived step 1. Let $F_1,F_2...,F_n$ be their cdf(cumulative distribution function) and let $f_1,f_2...,f_n$ be their pdf(probability density function). Let $[r_i,\:s_i]$ be the interval of $X_i$. Let $r_k$ be the biggest of all $r_i$.
I suggest a reduction phase. Remove all random variables $X_i$ that have $s_i<r_k$. Those cannot be the maximum.
After this step of reducing the problem to a subset of the random variables, we can start looking at the probability aspect of this.
Now the probability that the following holds $X_m = \max\{X_1, X_2, X_3,\ldots, X_n\}$ amounts to:
$$\int_{r_k}^{s_m}\: f_m(x)\prod_{i=1,i\neq m}^n F_i(x) \; dx$$
$$=\frac{1}{s_m-r_m}\int_{r_k}^{s_m}\: \prod_{i=1,i\neq m}^n F_i(x) \; dx$$
For uniform distributed random variables we know their cdf, namely:
$$F_i(x)=\left\{   
\begin{array}{ll} 
0  & x<r_i \\
\frac{x-r_i}{s_i-r_i} &  r_i \leq x < s_i \\ 
1 & x \geq s_i   
\end{array} 
\right.$$
Or equivalently: 
$$F_i(x)=\left\{   
\begin{array}{ll} 
0  & x<r_i \\
\frac{1}{s_i-r_i}(\min(x,s_i)-r_i) &  r_i \leq x     
\end{array} 
\right.$$
Using the fact that we choose $r_k$ to be the biggest of all lower bounds of the intervals, we only have case 2 for our cdf. This is the equation you get :
$$\prod_{i=1}^n{\frac{1}{s_i-r_i}}\int_{r_k}^{s_m}\: \prod_{i=1,i\neq m}^n (\min(x,s_i)-r_i)  \; dx$$
Now some the knowledge of the interval can be used. Let the upper bounds of the intervals $s_i$ be increasing or equal if $i$ increases, this can always be done by changing the index of each random variable. Now cutting the integration area in parts we get:
$$\int_{r_k}^{s_m}\: \prod_{i=1,i\neq m}^n (\min(x,s_i)-r_i)  \; dx=$$
$$\int_{r_k}^{s_1}\: \prod_{i=1,i\neq m}^n (x-r_i)  \; dx + \sum_{j=1}^{m-1}\int_{s_j}^{s_{j+1}}\: \prod_{i=j+1,i\neq m}^n (x-r_i)  \; dx=$$
Now you have all integrals of polynomials of degree at most n, this can be computed.
I reckon this won't have a lovely runtime in the number of variables left after the reduction, because a random variable makes the degree of all polynomials one higher in the worst case.
I hope this helps.
