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 -[itex]\infty[/itex] and [itex]\infty[/itex] 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!