3 Properties of Distributions
3.1 Parameters of Univariate Distributions
meristic characters (a range of discrete classes)
metric characters (continuous scale)
all-or-none or binary characters
univariate distribution
bivariate distribution
multivariate distribution
pdf: probability density function
pmf: probability mass function
parameters
estimates
Statisticians often denote parameters of a population with Greek symbols and to sample estimates with Roman symbols
arithmetic mean, first moment about the origin, expected value, expectation
population variance, second moment about the mean
standard deviation (SD)
coefficient of variation (the ratio of the standard deviation to the mean)
third moment about the mean: asymmetry of a distribution, skewness
coefficient of skewness (a ratio)
3.2 The Normal Distribution
Normal distribution/Gaussian distribution (DeMoivre(1738), LaPlace(1778), and Gauss(1809)), the density function is given by
\[p(z) = (2\pi\sigma^2)^{-1/2}exp[-\frac{(z-\mu)^2}{2\sigma^2}]\]
central limit theorem
Gaussian fitness function (in the theory of stablizing selection)
standard normal distribution
third moment is 0
fourth moment, kurtosis, leptokurtic vs platykurtic
3.2.1 The truncated normal distribution
Under truncation selection, all individuals below a certain phenotype are culled from the population and hence have zero fitness. The critical phenotype, T, is called the truncation point. For a normally distributed pheontype, the density of the phenotype z after selection is
\[\frac{p(z)}{\int_T^\infty p(z)dz}\] , the mean phenotype of the population above the threshold (i.e., after selection) can be writen as
\[\mu_s = \frac{\int_T^\infty zp(z)dz}{\int_T^\infty p(z)dz} = \mu+\frac{\sigma \times p_T}{\Phi_T}\]
, where
- \(\Phi_T\) is the denominator \(\int_T^\infty p(z)dz\), a measure of the intensity of selection;
- \(p_T\) is the height of the standard normal curve at the truncation point T.
The change in the mean caused by selection (\(\mu_s-\mu = \frac{\sigma \times p_T}{\Phi_T}\)) is often denoted by S, the directional selection differential.
In a similar fashion, the variance of the selected population can be shown to be
\[\sigma^2_s = [1+\frac{p_Tz'}{\Phi_T}-(\frac{p_T}{\Phi_T})^2]\sigma^2\] , where \(z' = T - \mu\)
mu = 0; sigma=1
phi_T = 10^(seq(-3,0,by=0.05))
Tvalue = qnorm(phi_T, lower.tail=F)
p_T = dnorm(Tvalue)
mu_s = mu+sigma*p_T/phi_T
layout(matrix(1:2, 1, 2))
plot(x=log10(phi_T), y=(mu_s-mu)/sigma, type="l", xlab=expression(log[10]~Phi[T]), ylab=expression((mu[s]-mu)/sigma))
plot(x=log10(phi_T), y=(1+p_T*(Tvalue-mu)/phi_T-(p_T/phi_T)^2), type="l", xlab=expression(log[10]~Phi[T]), ylab=expression(sigma[s]^2/sigma^2))