Statistical Inference Final Project Part 1

# task
# set seed for reproducability
set.seed(31)

# set lambda to 0.2
lambda <- 0.2

# 40 samples
n <- 40

# 1000 simulations
simulations <- 1000

# simulate
simulated_exponentials <- replicate(simulations, rexp(n, lambda))

# calculate mean of exponentials
means_exponentials <- apply(simulated_exponentials, 2, mean)

Question 1

# distrribution mean
analytical_mean <- mean(means_exponentials)
analytical_mean

## [1] 4.993867

# analytical mean
theory_mean <- 1/lambda
theory_mean

## [1] 5

# visualization
hist(means_exponentials, xlab = "mean", main = "Exponential Function Simulations")
abline(v = analytical_mean, col = "blue")
abline(v = theory_mean, col = "red")

Question 2

# standard deviation of distribution
standard_deviation_dist <- sd(means_exponentials)
standard_deviation_dist

## [1] 0.7931608

# standard deviation from analytical expression
standard_deviation_theory <- (1/lambda)/sqrt(n)
standard_deviation_theory

## [1] 0.7905694

# variance of distribution
variance_dist <- standard_deviation_dist^2
variance_dist

## [1] 0.6291041

# variance from analytical expression
variance_theory <- ((1/lambda)*(1/sqrt(n)))^2
variance_theory

## [1] 0.625

Question 3

xfit <- seq(min(means_exponentials), max(means_exponentials), length=100)
yfit <- dnorm(xfit, mean=1/lambda, sd=(1/lambda/sqrt(n)))
hist(means_exponentials,breaks=n,prob=T,col="blue",xlab = "means",main="Density of means",ylab="density")
lines(xfit, yfit, pch=22, col="black", lty=5)

# compare the distribution of averages of 40 exponentials to a normal distribution
qqnorm(means_exponentials)
qqline(means_exponentials, col = 2)