1. ## Sample mean PDF

I read a proof which showed that if X follows a normal distribution with mean $\mu$ and variance $\sigma^2$ then the sample mean also follows a normal distribution with mean $\mu$ and variance $\frac{\sigma^2}{n}$, 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: $P(y,\mu,\sigma^2)$ then the sample mean is distributed according to $P(\bar{y},\mu,\frac{\sigma^2}{n})$

2. ## Re: Sample mean PDF

Originally Posted by Shakarri
I read a proof which showed that if X follows a normal distribution with mean $\mu$ and variance $\sigma^2$ then the sample mean also follows a normal distribution with mean $\mu$ and variance $\frac{\sigma^2}{n}$, 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: $P(y,\mu,\sigma^2)$ then the sample mean is distributed according to $P(\bar{y},\mu,\frac{\sigma^2}{n})$
No, asymtotically for large $n$ the distribution is $N(\mu,\sigma^2/n)$ it's the Central Limit Theorem.

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