recentpopularlog in

nhaliday : integral   25

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
A brief philosophical discussion:
Measure theory, as much as any branch of mathematics, is an area where it is important to be acquainted with the basic notions and statements, but not desperately important to be acquainted with the detailed proofs, which are often rather unilluminating. One should always have in a mind a place where one could go and look if one ever did need to understand a proof: for me, that place is Rudin’s Real and Complex Analysis (Rudin’s “red book”).
gowers  pdf  math  math.CA  math.FA  philosophy  measure  exposition  synthesis  big-picture  hi-order-bits  ergodic  ground-up  summary  roadmap  mathtariat  proofs  nibble  unit  integral  zooming  p:whenever 
february 2017 by nhaliday
Cauchy-Schwarz inequality and Hölder's inequality - Mathematics Stack Exchange
- Cauchy-Schwarz (special case of Holder's inequality where p=q=1/2) implies Holder's inequality
- pith: define potential F(t) = int f^{pt} g^{q(1-t)}, show log F is midpoint-convex hence convex, then apply convexity between F(0) and F(1) for F(1/p) = ||fg||_1
q-n-a  overflow  math  estimate  proofs  ground-up  math.FA  inner-product  tidbits  norms  duality  nibble  integral 
january 2017 by nhaliday
measure theory - Continuous function a.e. - Mathematics Stack Exchange
- note: Riemann integrable iff continuous a.e. (see Wheeden-Zygmund 5.54)
- equal a.e. to continuous f, but not continuous a.e.: characteristic function of rationals
- continuous a.e., but not equal a.e. to continuous f: step function
- continuous a.e., w/ uncountably many discontinuities: characteristic function of Cantor set
q-n-a  overflow  math  math.CA  counterexample  list  measure  smoothness  singularity  nibble  integral 
january 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:

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

Copy this bookmark:

to read