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**Shakarri** I read a proof which showed that if X follows a normal distribution with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$ then the sample mean also follows a normal distribution with mean $\displaystyle \mu$ and variance $\displaystyle \frac{\sigma^2}{n}$, is this true in the general case so that if a random variable Y has a probability density function which is a function of mean, variance and y: $\displaystyle P(y,\mu,\sigma^2)$ then the sample mean is distributed according to $\displaystyle P(\bar{y},\mu,\frac{\sigma^2}{n})$