Let $X\_1, X\_2, ...X\_n$ be independent and uniformly distributed random variables on the interval $[0,1]$. Now suppose I wanted to calculate the probability density function of $Z = X\_1 + X\_2 + ... + X\_n$. I think this can be done by $n-1$ successive convolutions, but that's not too important for me right now. My confusion stems from the plot on the bottom which shows the resulting PDF's where $n = 2,3,4,5,6$. Obviously we no longer get a uniformly distributed random variable, but what's puzzling to me is the fact, that the new PDF has range $[0,n]$. This result only makes sense to me if we assume that the $X\_i$ actually all have the *same range* (in this case $0 \leq X\_i \leq 1$ for all $i$). Informally, what keeps $X\_1$ from being the amount of fuel in a passing car and say $X\_2$ the number of passengers in said car?

[![taken from the german Wikipedia article on the uniform dist. ](https://i.stack.imgur.com/ZT5rj.png)](https://i.stack.imgur.com/ZT5rj.png)