Positive probabilities for neg sums of uniformly distributed variables

[colinman]

I've been thinking about the Central Limit Theorem and by my understanding it states that the sum of randomly distributed variables follows approximately a normal distribution.

My question is if you have, say, 100 uniformly distributed variables that range from 0 to 10, their sum has to be positive (since probability is 0 everywhere outside of the min and max). However, CLT states that the sum follows a normal distribution centered around 500 (since the mean of each random variable is 5 and there are 100 of them).

A normal distribution assigns a positive probability to every value between -\infty and \infty even if it's a super small probability. Does this mean that the probability of the sum of the random variables being negative is not 0?

This has been bothering me for a while, so any input is appreciated. Thanks!

 

[Simon Bridge]

Welcome to PF;
You are correct, the P(x<0)=0 no matter how many terms there are to the sum.

You'll also notice that P(x=0)>0 too, where it should be exactly 0 for the normal distribution.

What you are getting here is an approximate Normal distribution that has been shifted to the right.
You have shown what happens for your example where N=100 ... what happens to your example as N approaches infinity?

 

[micromass]

I've been thinking about the Central Limit Theorem and by my understanding it states that the sum of randomly distributed variables follows approximately a normal distribution.

My question is if you have, say, 100 uniformly distributed variables that range from 0 to 10, their sum has to be positive (since probability is 0 everywhere outside of the min and max). However, CLT states that the sum follows a normal distribution centered around 500 (since the mean of each random variable is 5 and there are 100 of them).

A normal distribution assigns a positive probability to every value between -\infty and \infty even if it's a super small probability. Does this mean that the probability of the sum of the random variables being negative is not 0?

This has been bothering me for a while, so any input is appreciated. Thanks!

If you add up 100 variables, then the sum will not be normally distributed. It will only approximately be unormally distributed. In fact, if the original variables are uniformly distributed from 0 to 10, then the sum of the 100 variables will be approximately normally distributed with mean 500 and variance about $833\sim 29^2$.

Now, since we have only approximately normal distribution, then the probability that the uniform distribution be <0 is approximately equal to the probability that the normal distribution is <0. So it is not exactly equal.

Now, the probability that the sum of the uniform distributions be <0 is of course 0. The probability that the normal distribution is <0 is nonzero, but is so very close to 0 that many calculators will give it as equal to 0 (for example: http://davidmlane.com/hyperstat/z_table.html).

So the central limit theorem works in the sense that the probabilities are indeed very close to each other, but not equal.

 

[Stephen Tashi]

The central limit theorem doesn't say that the sum of independent random variables is approximately normally distributed. It says that a certain function of the sum is approximately normally distributed. In your example, this function of the sum can take on negative values even though the sum itself cannot.

 

[colinman]

Thank you so much for the responses! I think I understand it now :)