Probability Density Function of Average Value of Log-Normal Trials

[bbpr]

X is a random variable that follows the Log-Normal probability density function.
n indipendent trials are carried out.
We want to know the probability density function of the random variable Y, that is defined as the average value of the “n” outcomes of the trials described above.

 

[Jhero]

The probability density function of Y is given by the Central Limit Theorem, which states that for a sufficiently large number of trials, the distribution of the average of a random variable tends to a normal distribution with mean equal to the mean of the original distribution and variance equal to the variance of the original distribution divided by the number of trials.Therefore, in our case, the probability density function of Y is a Normal Distribution with mean equal to the mean of the Log-Normal distribution, and variance equal to the variance of the Log-Normal distribution divided by n.