This is the first part of the final project for the course Statistical Inference by JHU in Coursera.


In this project, we will simulate the exponential distribution and compare it with the Central Limit Theorem (CLT). As mentioned by Prof. Brian Caffo (2016), “CLT states that the distribution of averages of iid variables becomes that of a standard normal as the sample size increases”. The “standard normal” is that of a bell-curve shaped[1].

Taboga (2020) described exponential distribution as a continuous probability distribution used to model the time we need to wait before a given event occurs. This is pretty self-explanatory but to make an analogy out of it, imagine that this is the number of minutes left before a bomb explodes. So, distribution’s shape should be like a downward trend[2].

In this simulation, we want to see the difference between the theoretical mean and theoretical variance of an exponential distribution and its actual mean and actual variance when we’ve taken 40 samples from an exponential distribution, take their mean, and do that process by a thousand times.


Comparing Means

First, let’s define the theoretical mean and variance of an exponential distribution. We will use the R function rexp that takes two arguments. The first argument is the number of random number and the second argument is the rate or lambda (\(\lambda\)). Our \(\lambda\) in this study is 0.2. The mean and standard deviation of an exponential distribution is \(1 /\lambda\).

theo <- 1 / 0.2 
cat('Theoretical mean: ', theo, '\nTheoretical standard deviation: ', theo)
## Theoretical mean:  5 
## Theoretical standard deviation:  5


We can then directly create the histogram of 1 thousand simulations of 40 random samples from an exponential distribution.

set.seed(40000)
means <- NULL
for (i in 1:1000) means[i] <- mean(rexp(40, 0.2))
ggplot() + 
  geom_histogram(aes(x = means, y = ..density..), 
                 bins = 20, col = 'grey', fill = 'steelblue') + 
  labs(x = '1000 means of 40 samples from Exponential Distribition', 
       y = 'Count', title = 'Exponential Distribution on CLT') +
  geom_vline(xintercept = theo, color = 'red', size = 1) +
  geom_vline(xintercept = mean(means), color = 'orange', 
             size = 1.5, linetype = 'dotted') + 
  theme_minimal()

The above histogram shows us the means of 40 random samples from an exponential distribution which is done 1000 times. As we can see, the shape is close to a bell-curve that is used to portray a normal distribution. Looking at the values, the actual mean (orange dotted line) 5.05 is so close to the theoretical mean (red line) of 5.


Comparing Variances

Let’s now look on the theoretical and actual variances.

theo_var <- (1 / 0.2) ^ 2 / 40
act_var <- var(means)
cat('Actual variance: ', act_var, '\ntheoretical variance', theo_var)
## Actual variance:  0.6524401 
## theoretical variance 0.625

The values are still close to each other. Let’s now visualize it.

xnorm <- seq(min(means), max(means), length = 1000)
ynorm <- dnorm(xnorm, mean = 5, sd = 1/0.2/sqrt(40))
ggplot() + 
  geom_area(aes(x = xnorm, y = ynorm), size = 1, col = 'red', 
            fill = 'red', alpha = 0.5) + 
  geom_density(aes(x = means), size = 1, col = 'steelblue', fill = 'steelblue', 
               alpha = 0.5) + 
  theme_minimal() +
  labs(x = 'Spread', y = 'Density', 
       title = 'Variances of theoretical and actual exponential curve under CLT')

We can see from above densities that the two are so close to each other which means that the two are not that too variable when compared with each other.


Distribution

Let’s now see if the distribution of our simulation is really close to a normal distribution.

ggplot() + 
  stat_qq(aes(sample = means), col = 'steelblue', size = 1, alpha = 0.5) + 
  stat_qq_line(aes(sample = means), col = 'red', size = 1, alpha = 0.5) +
  labs(title = 'QQPlot Showing Approximate Normal Distribution') + 
  theme_minimal()

The above shows us that dots and the line somehow forms a line which means that it follows the same distribution.


References

Cafo, B. (2016). Statistical Inference for Data Science. Leanpub

Taboga, M. (2020). Exponential Distribution. StatLect. https://www.statlect.com/probability-distributions/exponential-distribution. Retrieved: December 12, 2020.


Appendix

Normal Distribution Curve


Exponential Distribution Curve