Central Limit Theorem - Shiny app

Definition

• Xi (i=1 to n) is a collection of iid random variables with mean mu and variance var and Sn the average

\[ S_{n}=\frac{X_{1} + X_{2}+ ... + X_{n}}{n} \]

For large n, in accordance with the Central Limit Theorem, the mean of iid random variables should follow a normal distribution with mean = mu and variance = var/n

Or for large n, \( \frac{\bar X_n - mu}{var / \sqrt{n}} \) has a distribution like that of a standard normal.

Synopsis

The purpose of the shiny application is to illustrate the Central Limit Theorem.
The user can choose :

• The number of iid random variables : nVar
• The number of observations to build the density : nObs
• The distribution : dist
  

See the application
https://yclaudel.shinyapps.io/firstshinyap

Calculation

• The server is going to calcultate n means.
• Each mean is the mean of nVar variables of the distribution dist.

mns = NULL
for (i in 1 : nObs) mns = c(mns, mean(dist(nVar)))
...
hist(mns)