Sortostat/Optimal sorting cutoffs

From OpenWetWare
Jump to: navigation, search

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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph E[X_N] = } expected value of the optimal percentage that can be returned from N trials

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph S_i = S} , for all i

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph C_i = } the cut-off percentile for the ith trial.


General

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph E[X_N] = P(S>C_1) E[S|S>C_1] + }


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph (1-P(S>C_1))(P(S>C_2)E[S|S>C_2]) + }


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph (1-P(S>C_1))(1-P(S>C_2))(P(S>C_3)E[S|S>C_2]) + }


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \emph ... }


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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.

Results

Contact

Jason Kelly