Set the simulation variables lambda, exponentials, and seed.

# set seed for reproducability
set.seed(2016)

# Set sampling values as described in the project instructions
lambda <- 0.2   # lambda
n <- 40         # number of exponentials
sims <- 1000    # number of simulations

#Run simulations
sim_exp <- replicate(sims, rexp(n, lambda))

#Calc the means of the exponential simulations
means_exp <- apply(sim_exp, 2, mean)

#Histogram of the means
hist(means_exp, breaks=40, xlim = c(2,9), main="Exponential Function Simulation Means", col = "aliceblue")

plot(rexp(10000,lambda), pch=20, cex=0.6, main="The exponential distribution with rate 0.2 and 10.000 observations")

Question 1 - Sample Mean vs Theoretical Mean

The mean of the exponential distribution is 1/lambda. In this case, lambda is 0.2. Therefore, the theoretical mean should result as 5 (i.e. 1 / 0.2). Let’s see if that holds true.

# plot histogram of the sample means
hist(means_exp, col="aliceblue", main="Theoretical Mean vs. Actual Mean", xlim = c(2,9),breaks=40, xlab = "Simulation Means")

# plot a vertical red line at the mean of the sample means
abline(v=mean(means_exp), lwd="4", col="red")

# determine the mean of our sample means
mean(means_exp)

Question 2 - Sample Variance vs Theoretical Variance

The standard deviation of the exponential distribution is (1/lambda) / sqrt(n). Next, we’ll see if this matches our simulations.

# theoretical standard deviation vs. simulation standard deviation
print(paste("Theoretical standard deviation: ", round( (1/lambda)/sqrt(n) ,4)))
print(paste("Practical standard deviation: ", round(sd(means_exp) ,4))) 
print(paste("Theoretical variance: ", round( ((1/lambda)/sqrt(n))^2 ,4)))
print(paste("Practical variance: ", round(sd(means_exp)^2 ,4))) 
hist(means_exp, col="darkblue", main="Theoretical vs actual mean for rexp()", breaks=20)
abline(v=mean(means_exp), lwd="4", col="red")

The formulas above show us that the variances are very close.

Question 3 - Distribution

Finally, we’ll investigate whether the exponential distribution is approximately normal. Due to the Central Limit Theorem, the means of the sample simulations should follow a normal distribution.

#General Plot with ditribution curve drawn
hist(means_exp, prob=TRUE, col="lightblue", main="mean distribution for rexp()", breaks=20)
lines(density(means_exp), lwd=3, col="blue")

Statistical Inference Course Project

Alfredo Aranda N.

5 de febrero de 2018

Set the simulation variables lambda, exponentials, and seed.

Question 1 - Sample Mean vs Theoretical Mean

The mean of the exponential distribution is 1/lambda. In this case, lambda is 0.2. Therefore, the theoretical mean should result as 5 (i.e. 1 / 0.2). Let’s see if that holds true.

Question 2 - Sample Variance vs Theoretical Variance

The standard deviation of the exponential distribution is (1/lambda) / sqrt(n). Next, we’ll see if this matches our simulations.

The formulas above show us that the variances are very close.

Question 3 - Distribution

Finally, we’ll investigate whether the exponential distribution is approximately normal. Due to the Central Limit Theorem, the means of the sample simulations should follow a normal distribution.