Central Limit Theorem and Uniform Distribution

Question

How does central limit theorem hold good for uniform distribution? Suppose $p(x)$ for $x=1,2,3…,n$ be $1/n$. If we take a sample say $1,2,3$ the mean is $2$ but since it is uniform the probability value will be $1/n$ isn’t it? And the same for all samples, how does this look a normal curve when the number of samples increase?

Answer 1

Let's look at what happens when everything is a uniform distribution on $\{0, 1\}$.

If there's just one, then the possible values are 0 and 1, and they're all equally likely, so $P(U_1 = 0) = P(U_1 = 1) = \frac{1}{2}$.

If you have two variables, then the possible values of their sum are in the range of 0 to 2, but they aren't evenly distributed because $U_1 + U_2 = 2$ only if $U_1 = U_2 = 1$ so it has a probability of $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$, but $U_1 + U_2 = 1$ can happen in 2 different ways - 0+1 and 1+0, so it has a probability of $2 \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{2}$.

So the probabilities become:

$u$	Ways to sum	$P(U_1 + U_2 = u)$
0	0+0	$\frac{1}{4}$
1	0+1, 1+0	$\frac{2}{4}$
2	1+1	$\frac{1}{4}$

When you get to 3 variables, it looks like this:

$u$	Ways to sum	$P(U_1 + U_2 + U_3 = u)$
0	0+0+0	$\frac{1}{8}$
1	0+0+1, 0+1+0, 1+0+0	$\frac{3}{8}$
2	0+1+1, 1+0+1, 1+1+0	$\frac{3}{8}$
3	1+1+1	$\frac{1}{8}$

You can see what happens for even more variables by playing with this Binomial Distribution visualisation, set "Probability of success" to 0.5 and adjust "Number of trials". When you increase that value, you'll see it become more and more bell-curve-like.