nhaliday + amt   20

Atrocity statistics from the Roman Era
Christian Martyrs [make link]
Gibbon, Decline & Fall v.2 ch.XVI: < 2,000 k. under Roman persecution.
Ludwig Hertling ("Die Zahl de Märtyrer bis 313", 1944) estimated 100,000 Christians killed between 30 and 313 CE. (cited -- unfavorably -- by David Henige, Numbers From Nowhere, 1998)
Catholic Encyclopedia, "Martyr": number of Christian martyrs under the Romans unknown, unknowable. Origen says not many. Eusebius says thousands.

...

General population decline during The Fall of Rome: 7,000,000 [make link]
- Colin McEvedy, The New Penguin Atlas of Medieval History (1992)
- From 2nd Century CE to 4th Century CE: Empire's population declined from 45M to 36M [i.e. 9M]
- From 400 CE to 600 CE: Empire's population declined by 20% [i.e. 7.2M]
- Paul Bairoch, Cities and economic development: from the dawn of history to the present, p.111
- "The population of Europe except Russia, then, having apparently reached a high point of some 40-55 million people by the start of the third century [ca.200 C.E.], seems to have fallen by the year 500 to about 30-40 million, bottoming out at about 20-35 million around 600." [i.e. ca.20M]
- Francois Crouzet, A History of the European Economy, 1000-2000 (University Press of Virginia: 2001) p.1.
- "The population of Europe (west of the Urals) in c. AD 200 has been estimated at 36 million; by 600, it had fallen to 26 million; another estimate (excluding ‘Russia’) gives a more drastic fall, from 44 to 22 million." [i.e. 10M or 22M]

also:
The geometric mean of these two extremes would come to 4½ per day, which is a credible daily rate for the really bad years.

why geometric mean? can you get it as the MLE given min{X1, ..., Xn} and max{X1, ..., Xn} for {X_i} iid Poissons? some kinda limit? think it might just be a rule of thumb.

yeah, it's a rule of thumb. found it it his book (epub).
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september 2017 by nhaliday
Beta function - Wikipedia
B(x, y) = int_0^1 t^{x-1}(1-t)^{y-1} dt = Γ(x)Γ(y)/Γ(x+y)
one misc. application: calculating pdf of Erlang distribution (sum of iid exponential r.v.s)
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march 2017 by nhaliday
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

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