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hyginn 2017-10-12 15:14:46 -04:00
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FND-STA-Probability_distribution.R
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@ -21,9 +21,9 @@
# going on. That's not how it works ... # going on. That's not how it works ...
# #
# ============================================================================== # ==============================================================================
#TOC> ========================================================================== #TOC> ==========================================================================
#TOC> #TOC>
#TOC> Section Title Line #TOC> Section Title Line
#TOC> ----------------------------------------------------------------------- #TOC> -----------------------------------------------------------------------
#TOC> 1 Introduction 50 #TOC> 1 Introduction 50
@ -40,9 +40,9 @@
#TOC> 4.2.1 An example from tossing dice 446 #TOC> 4.2.1 An example from tossing dice 446
#TOC> 4.2.2 An example from lognormal distributions 568 #TOC> 4.2.2 An example from lognormal distributions 568
#TOC> 4.3 Kolmogorov-Smirnov test for continuous distributions 609 #TOC> 4.3 Kolmogorov-Smirnov test for continuous distributions 609
#TOC> #TOC>
#TOC> ========================================================================== #TOC> ==========================================================================
@ -58,26 +58,26 @@
# Let's get a few facts about probability distributions out of the way: # Let's get a few facts about probability distributions out of the way:
# The "support" of a probability distribution is the range of values that have a # The "support" of a probability distribution is the range of outcomes that have
# non-zero probability. The "domain" of a probability distribution is the range # a non-zero probability. The "domain" of a probability distribution is the
# of probabilities that the distribution can take over its support. Think of # range of probabilities that the distribution can take over its support. Think
# this as the ranges on the x- and y-axis respectively. Thus the distribution # of this as the ranges on the x- and y-axis respectively. Thus the distribution
# can be written as p = f(x). # can be written as p = f(x).
# The integral over a probability distribution is always 1. This means: the # The integral over a probability distribution is always 1. This means: the
# distribution reflects the situation that an event does occur, any event, but # distribution reflects the situation that an event does occur, any event, but
# there is not no event. # there is not "no event".
# R's inbuilt probability distributions always come in four flavours: # R's inbuilt probability functions always come in four flavours:
# d... for "density": this is the probability density function (p. d. f), # d... for "density": this is the probability density function (p.d.f.),
# the value of f(x) at x. # the value of f(x) at x.
# p... for "probability": this is the cumulative distribution function # p... for "probability": this is the cumulative distribution function
# (c. d. f.). It is 0 at the left edge of the support, and 1 at # (c.d.f.). It is 0 at the left edge of the support, and 1 at
# the right edge. # the right edge.
# q... for "quantile": The quantile function return the x value at which p... # q... for "quantile": The quantile function returns the x value at which p...
# takes a requested value. # takes a requested value.
# r... for "random": produces random numbers that are distributed according # r... for "random": produces random numbers that are distributed according
# to the p. d. f. # to the p.d.f.
# To illustrate with the "Normal Distribution" (Gaussian distribution): # To illustrate with the "Normal Distribution" (Gaussian distribution):
@ -443,7 +443,7 @@ chisq.test(countsL1, countsG1.9, simulate.p.value = TRUE, B = 10000)
# be applied to discrete distributions. But we need to talk a bit about # be applied to discrete distributions. But we need to talk a bit about
# converting counts to p.m.f.'s. # converting counts to p.m.f.'s.
# === 4.2.1 An example from tossing dice # === 4.2.1 An example from tossing dice
# The p.m.f of an honest die is (1:1/6, 2:1/6, 3:1/6, 4:1/6, 5:1/6, 6:1/6). But # The p.m.f of an honest die is (1:1/6, 2:1/6, 3:1/6, 4:1/6, 5:1/6, 6:1/6). But
# there is an issue when we convert sampled counts to frequencies, and estimate # there is an issue when we convert sampled counts to frequencies, and estimate
@ -565,7 +565,7 @@ abline(v = KLdiv(rep(1/6, 6), pmfPC(counts, 1:6)), col="firebrick")
# somewhat but not drastically atypical. # somewhat but not drastically atypical.
# === 4.2.2 An example from lognormal distributions # === 4.2.2 An example from lognormal distributions
# We had compared a set of lognormal and gamma distributions above, now we # We had compared a set of lognormal and gamma distributions above, now we
# can use KL-divergence to quantify their similarity: # can use KL-divergence to quantify their similarity: