Properties of Distributions

  • univariate/bivariate/multivariate distribution
  • probability distributions
  • probability density function \(p(z)\)

\[P(z_{1}\leq z \leq z_{2})=\int_{z_{1}}^{z_{2}}p(x)dz\]

What is moment about the origin?

Moments. The moments of a distribution are a set of parameters that summarize it. Given a random variable \(X\), its first moment about the origin, denoted , is defined to be \(E[X]\). Its second moment about the origin is the population variance.

eg.

# normal distribution
integrate(dnorm, -1.96, 1.96)
## 0.9500042 with absolute error < 1e-11
# negative exponential distribution
integrate(function(x) 2*exp(-2*x), 1/4, 1/2)
## 0.2386512 with absolute error < 2.6e-15
# expectation
integrate(function(x) x*2*exp(-2*x), 0, Inf)
## 0.5 with absolute error < 8.6e-06

normal distribution

\[p(z)=(2\pi\sigma^2)^{-1/2}exp[-\frac{(z-\mu)^2}{2\sigma^2}]\]

curve(dnorm(x),
      xlim = c(-3.5, 3.5),
      ylab = "Density", 
      main = "Standard Normal Density Function")

central limit theorem

truncated normal distribution

plant and animal breeding, truncation selection

library(truncnorm)
norm_data <- rnorm(n = 1000, mean = 90, sd = 4)
hist(norm_data)

truncnorm_data <- rtruncnorm(n = 1000, a = 85, b = 100, mean = 90, sd = 4)
mean(truncnorm_data)
## [1] 90.76805
sd(truncnorm_data)
## [1] 3.284828
hist(truncnorm_data)

confidence intervals