Dear members,

I'm a bit confused about a simple problem with log-normal distribution.

Here is the setup :

Let M be the true mean of some measurement on a population.

We do our measurement and obtain different Yi = Xi*Ei where Xi is the measurement of each individual and Ei is some error on the measurement.

The Ei follow a log-normal distribution with mean 0 and std 0.1

The Xi are first assumed to all have an identical value X. Then we tackle the case where they are log-normal with unknown parameters mu and sigma.

The fold-change in the measurement is defined as C= log(X/F) for identical Xi, and C = log(e^mu / F) for log-normal Xi. The empirical estimator is C_hat= log(Yi/F)/n.

What I managed was to say that for identical Xi, log(Yi/F) = log(Ei*Xi/F). log(Xi) - log(F) = 0, so we end up with only log(Ei). So log(Yi/F) is normally distributed with the same mean and std as the Ei (0,0.1). C_hat is also normally distributed with the same mean and std = sigma/sqrt(n)

I'm more hesitant as to what distribution the log(Yi/F) and C_hat follow for the other case. We have log(Yi/F) = log(Ei) + log(Xi) -log(F).

The log(Xi) and log(Ei) are normally distributed so I could just say the parameters are the sum of their mean, and the sum of their variance. But I'm unsure what to do with the log(F). Is log(F) equal to the unknown mean mu of the log(Xi) ? In which case I could say this centers the log(Xi) on mean = 0. This doesn't seem right, though, because F is the mean of the Xi, whereas mu is the mean of the log(Xi).

As seen on wikipedia :

where m would be F in this case if I'm correct.

So I am unsure of the distribution and parameters of the log(Yi/F), and of the distribution of C_hat and its parameters and then which test to apply if I wanted to compare C and C_hat.

Thank you very much for your help !

= 1

,...,n