I'm confused by the statement of the central limit theorem we've been given:
'If you take the sum $X$ of $N$ independent variables $x_i$, each taken from a distribution with mean $\mu_i$, then the distribution for $X$ has the average $\Sigma\mu_i$.'
So does this mean that if I take one number $x_i$ from a distribution and repeat the process $N$ times, then sum them I'll get a distribution that has mean $\Sigma\mu_i$? Where $\mu_i$ are the means of the distributions each $x_i$ has come from?
Or is it saying that if I take a sample $x_i$ each containing multiple numbers and calculate the mean of each sample to be $\mu_i$, then the distribution formed by the sample means has mean $\Sigma\mu_i$? Or something completely different?
I think what particularly confuses me is the idea that the mean of the new distribution $X$ should be a sum of means, because then won't your mean get bigger the more distributions you gather the $x_i$ from? But if all the distributions have a mean of 0.5 say, why should the distribution $X$ have a mean of $0.5N$? Our lecturer did show us that $$<X>=<\Sigma x_i>=\Sigma<x_i>=\Sigma\mu_i$$ But I can't get my head around the result.