# Sortostat/Optimal sorting cutoffs

## Problem

What is the optimal cut-off percentile for choosing a chamber to be sorted if you have N sorts (trials) remaining until you must take the sort to preserve a constant dilution rate?

## Analytical Solution

### Definition of variables

$\displaystyle{ \emph E[X_N] = }$ expected value of the optimal percentage that can be returned from N trials

$\displaystyle{ \emph S_i = }$ random variable representing the percentile returned from the ith trial

• all trials are assumed to be independent therefore $\displaystyle{ \emph S_i = S }$, for all i

$\displaystyle{ \emph C_i = }$ the cut-off percentile for the ith trial.

### General

$\displaystyle{ \emph E[X_N] = P(S\gt C_1) E[S|S\gt C_1] + }$

$\displaystyle{ \emph (1-P(S\gt C_1))(P(S\gt C_2)E[S|S\gt C_2]) + }$

$\displaystyle{ \emph (1-P(S\gt C_1))(1-P(S\gt C_2))(P(S\gt C_3)E[S|S\gt C_2]) + }$

$\displaystyle{ \emph ... }$

$\displaystyle{ \emph (1-P(S\gt C_1))(1-P(S\gt C_2))...(1-P(S\gt C_{N-1})E[S_N] }$

### Simplified

Since $\displaystyle{ \emph (1-P(S\gt C_1)) }$ can be factored out of every term after the first above, the solution can be simplified and solved recursively:

$\displaystyle{ \emph E[X_N] = P(S\gt C_N) E[S|S\gt C_N] + (1-P(S\gt C_N))E[X_{N-1}] }$

base case:

$\displaystyle{ \emph E[X_1] = \int_0^\infty P(S)*S dS }$

• e.g., if you have only 1 trial then you expect to get the mean of the distribution for S.

## Simulation Solution

Since our probability skills were pretty sad, we (Alex Mallet) simulated it to confirm our analytical results. MATLAB file can be found here.