# Solved – Mean and variance of multiple trials from normal dist

Imagine I have some process with mean \$mu\$ and variance \$sigma^2\$, which are both known empirically. If I sample from this process 1000 times, what's the probability that the mean of those 1000 samples is less than \$mu+epsilon\$?

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I have a binary classification model (returns probabilities using logistic regression). I can estimate the empirical mean log-loss and the variance of this log-loss, \$mu\$ and \$sigma^2\$ respectively. Let's say these are \$mu = 0.692\$ and \$sigma=0.01\$. If I then use my classifier on 1000 new sample points, I'd like to know the probability of my mean log-loss of my classifier across those samples being less than 0.693.

At the moment I have a pretty clumsy numerical method using the binomial distribution. I compute the CDF for a normal distribution using \$mu\$ and \$sigma\$ above to find the probability of any one point having log loss less than 0.693, then I sample the binomial distribution with this probability and aggregate the times when more than half the samples are below 0.693.

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