Computational Statistics

Chapter 3 - Methods for Generating Random Variables

Dr. Mehdi Maadooliat

Marquette University
MATH 4750 - Spring 2025

Lecture 1: Sampling and Permutations (Part 1)

Sampling from a Finite Population

  • Introduction to finite population sampling
  • Example: Tossing coins, choosing lottery numbers
# Sampling examples
sample(0:1, size = 10, replace = TRUE)  # Tossing coins
 [1] 1 1 1 0 1 1 1 1 0 1
sample(1:100, size = 6, replace = FALSE)  # Choosing lottery numbers
[1] 93 45 68 52 12 77
sample(letters)  # Permutation of letters
 [1] "f" "m" "o" "y" "j" "b" "g" "h" "x" "q" "l" "s" "u" "i" "d" "p" "n" "z" "a"
[20] "v" "e" "t" "k" "w" "c" "r"

Multinomial Distribution

  • Introduction to multinomial distribution
  • Example: Sampling from a multinomial distribution
# Sample from multinomial distribution
x <- sample(1:3, size = 100, replace = TRUE, prob = c(.2, .3, .5))
table(x)
x
 1  2  3 
17 36 47

Lecture 2: Inverse Transform Method (Part 2)

Continuous Case

  • Explanation of the inverse transform method for continuous distributions
  • Example: Simulating exponential random variables
# Inverse transform method for exponential distribution
n <- 1000
U <- runif(n)
X <- -log(1 - U)
hist(X, main="Exponential Distribution via Inverse Transform", col="lightblue")

Discrete Case

  • Example of applying the inverse transform method to discrete random variables
# Inverse transform method for geometric distribution
p <- 0.5
X_geom <- ceiling(log(1 - runif(n)) / log(1 - p))
hist(X_geom, main="Geometric Distribution via Inverse Transform", col="lightgreen")

Lecture 3: Acceptance-Rejection Method (Part 3)

Introduction to Acceptance-Rejection Method

  • Explanation of the acceptance-rejection method
  • Example: Sampling from a target distribution
# Acceptance-rejection method example
target <- function(x) { ifelse(x > 0, exp(-x), 0) }
proposal <- function(x) { dnorm(x, mean = 2, sd = 1) }

X <- rnorm(1000, mean = 2, sd = 1)
accept <- runif(1000) < target(X) / (1.5 * proposal(X))

hist(X[accept], main="Accepted Samples from Target Distribution", col="lightcoral")

Lecture 4: Transformation Methods and Mixtures (Part 4)

Transformation of Random Variables

  • Introduction to transformation methods
  • Example: Box-Muller transform for generating normal random variables
# Box-Muller transform
n <- 1000
U1 <- runif(n)
U2 <- runif(n)
Z1 <- sqrt(-2 * log(U1)) * cos(2 * pi * U2)
Z2 <- sqrt(-2 * log(U1)) * sin(2 * pi * U2)
hist(Z1, main="Normal Distribution via Box-Muller", col="lightblue")

Sums and Mixtures

  • Explanation of sums and mixtures of random variables
  • Example: Mixture of normals
# Mixture of normals
library(MASS)
mu1 <- 0; mu2 <- 3; sigma1 <- 1; sigma2 <- 2
p <- 0.3
X <- ifelse(runif(n) < p, rnorm(n, mu1, sigma1), rnorm(n, mu2, sigma2))
hist(X, main="Mixture of Normal Distributions", col="lightgreen")

Conclusion

  • Recap of random variable generation methods: sampling, inverse transform, acceptance-rejection, and transformations
  • Practice problems for next lecture

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