nhaliday + acm + identity   18

Section 10 Chi-squared goodness-of-fit test.
- pf that chi-squared statistic for Pearson's test (multinomial goodness-of-fit) actually has chi-squared distribution asymptotically
- the gotcha: terms Z_j in sum aren't independent
- solution:
- compute the covariance matrix of the terms to be E[Z_iZ_j] = -sqrt(p_ip_j)
- note that an equivalent way of sampling the Z_j is to take a random standard Gaussian and project onto the plane orthogonal to (sqrt(p_1), sqrt(p_2), ..., sqrt(p_r))
- that is equivalent to just sampling a Gaussian w/ 1 less dimension (hence df=r-1)
pdf  nibble  lecture-notes  mit  stats  hypothesis-testing  acm  probability  methodology  proofs  iidness  distribution  limits  identity  direction  lifts-projections 
october 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)
concept  atoms  acm  math  calculation  integral  wiki  reference  identity  AMT  distribution  multiplicative 
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
gnxp  explanation  concept  bio  genetics  population-genetics  agri-mindset  analysis  scitariat  org:sci  nibble  methodology  distribution  tidbits  probability  stats  acm  AMT  limits  magnitude  identity  integral  street-fighting  symmetry  s:*  tails  multi  q-n-a  overflow  wiki  reference  objektbuch  proofs 
december 2016 by nhaliday

bundles : academeacmframemathmetaproblem-solving

related tags

accuracy  acm  agri-mindset  algorithms  amortization-potential  AMT  analysis  approximation  arrows  atoms  bias-variance  bio  bits  calculation  characterization  cheatsheet  coding-theory  composition-decomposition  concept  definition  differential  direction  distribution  duality  entropy-like  examples  explanation  exposition  extrema  fourier  genetics  gnxp  ground-up  hypothesis-testing  identity  IEEE  iidness  information-theory  integral  lecture-notes  levers  lifts-projections  limits  linear-algebra  linearity  list  magnitude  marginal  martingale  math  math.CA  math.CV  mathtariat  metabuch  methodology  metrics  mit  ML-MAP-E  moments  multi  multiplicative  nibble  nitty-gritty  objektbuch  ocw  online-learning  optimization  ORFE  org:bleg  org:sci  overflow  pdf  physics  pic  plots  population-genetics  probability  proofs  properties  q-n-a  qra  reduction  reference  rigor  s:*  scitariat  sequential  series  simplex  stats  stochastic-processes  street-fighting  symmetry  tails  tcs  tidbits  visual-understanding  wiki  yoga 

Copy this bookmark: