Stem Construction Utility

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Core component of the stem style P v replicate k 1 : i 1 v length range P i el 1 ≡ continue j 1 P i el range T k j + el v i el : for k k P i el + : 21 line if for T 31 line :

Stem construction vy stem N vy length : i 1 N range q i el 3 : q N el 2 : 21 line for 11 line i 1 N range v i el i : for i 1 N range S 3 i * el vy i el : for u q v replicate : V u S augment : V V 3 V rows 12 submatrix : 71 line :

E.g. for a logistic distribution with parameters m and b, the cumulative distribution function is: F(x) = 1/(1+exp(-(x-m)/b)) ... so the inverse is G(x) = m - b*log(1/y - 1) A random sample can then be drawn with gen y = `m' - `b'*log(1/uniform() - 1) (where `m' and `b' are locals containing the parameters m and b)

x m b q 11 x m - (- b / exp + / :

y x m b q ≡ x solve maple m 1 - y + y / ln b * -

# uniform size 1100 Random.N ≡

L 1 : S 0.5 : 12 mat

Number of individual samples to collect
Number of data points in each sample NSamples 50 : SSize 200 : 21 sys

a 0.95 : b 1.05 : 12 mat

Means λ 10 : count 1 : i 1 NSamples range xRnd 2 λ / SSize 1 λ Random.N * : SSize L S rlogis L 1 xRnd / ln vectorize S * - : rnd SSize L S rlogis : LocalLogis count el rnd Mean b a - 2 / + : count count 1 + : 51 line for LocalLogis 41 line :

Means Means :

Means Min Means Max 21 mat 0.945 1.066 21 mat

Success a Means i el ≤ (Means i el b ≤ (& i 1 NSamples sum :

Prob Success NSamples / : 0.880

Random Means stem :

Monte Carlo Probability Estimation

Can estimate probabilities of quantities whose distributions

are unknown. Estimation of a sample mean of a logistic

distribution is used as an example. More formal methods

should be considered from advanced tools . . . R

This Smath document is demo only.

Determine empirically the probability that the sample mean,

based on a sample of size NSamples of a logistic distribution

with parameters L and S will lie in the interval [a, b]

]]> b a Random 3 1 sys

Prob 0.88

MCD