Sortostat/Optimal sorting cutoffs
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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
![\emph E[X_N] =](/images/math/d/0/5/d0505d429341361b0dcfe2117051559b.png)
expected value of the optimal percentage that can be returned from N trials
random variable representing the percentile returned from the ith trial
- all trials are assumed to be independent therefore
, for all i
the cut-off percentile for the ith trial.
General
![\emph E[X_N] = P(S>C_1) E[S|S>C_1] +](/images/math/2/1/0/21083d43e8948db4c38302b8db549d78.png)
![\emph (1-P(S>C_1))(P(S>C_2)E[S|S>C_2]) +](/images/math/b/4/2/b429e14462bbca7fabe0861383db31f4.png)
![\emph (1-P(S>C_1))(1-P(S>C_2))(P(S>C_3)E[S|S>C_2]) +](/images/math/2/9/2/292baa87778d98e660465ce58046c4b8.png)
![\emph (1-P(S>C_1))(1-P(S>C_2))...(1-P(S>C_{N-1})E[S_N]](/images/math/8/1/f/81fd90835453aea76155c082c4631c88.png)
Simplified
Since 
can be factored out of every term after the first above, the solution can be simplified and solved recursively:
![\emph E[X_N] = P(S>C_N) E[S|S>C_N] + (1-P(S>C_N))E[X_{N-1}]](/images/math/9/3/2/9329b15c0e2e7b7079796aeda075679f.png)
base case:
![\emph E[X_1] = \int_0^\infty P(S)*S dS](/images/math/6/9/d/69d75d866c29fd05b9f2778a2c111ef6.png)
- 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.
