nhaliday + acm + population-genetics   5

Breeding the breeder's equation - Gene Expression
- interesting fact about normal distribution: when thresholding Gaussian r.v. X ~ N(0, σ^2) at X > 0, the new mean μ_s satisfies μ_s = pdf(X,t)/(1-cdf(X,t)) σ^2
- follows from direct calculation (any deeper reason?)
- note (using Taylor/asymptotic expansion of complementary error function) that this is Θ(t) as t -> 0 or ∞ (w/ different constants)
- for X ~ N(0, 1), can calculate 0 = cdf(X, t)μ_<t + (1-cdf(X, t))μ_>t => μ_<t = -pdf(X, t)/cdf(X, t)
- this declines quickly w/ t (like e^{-t^2/2}). as t -> 0, it goes like -sqrt(2/pi) + higher-order terms ~ -0.8.

Average of a tail of a normal distribution: https://stats.stackexchange.com/questions/26805/average-of-a-tail-of-a-normal-distribution

Truncated normal distribution: https://en.wikipedia.org/wiki/Truncated_normal_distribution
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december 2016 by nhaliday