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

Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {E}}[X_{N}]=} expected value of the optimal percentage that can be returned from N trials

Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {S}}_{i}=} random variable representing the percentile returned from the ith trial

  • all trials are assumed to be independent therefore Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {S}}_{i}=S} , for all i

Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {C}}_{i}=} the cut-off percentile for the ith trial.


General

Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {E}}[X_{N}]=P(S>C_{1})E[S|S>C_{1}]+}


Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {(}}1-P(S>C_{1}))(P(S>C_{2})E[S|S>C_{2}])+}


Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {(}}1-P(S>C_{1}))(1-P(S>C_{2}))(P(S>C_{3})E[S|S>C_{2}])+}


Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {.}}..}


Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {(}}1-P(S>C_{1}))(1-P(S>C_{2}))...(1-P(S>C_{N-1})E[S_{N}]}


Simplified

Since Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {(}}1-P(S>C_{1}))} can be factored out of every term after the first above, the solution can be simplified and solved recursively:

Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {E}}[X_{N}]=P(S>C_{N})E[S|S>C_{N}]+(1-P(S>C_{N}))E[X_{N-1}]}

base case:

Failed to parse (Conversion error. Server ("https://api.formulasearchengine.com/") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {E}}[X_{1}]=\int _{0}^{\infty }P(S)*SdS}

  • 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.

Results

Contact

Jason Kelly