Confusion about the range of the sum of i.i.d. random variables 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?

 A: You assumed the $X_i$ are uniformly distributed in $[0,1]$ in the first place, so why are you later puzzled that "the $X_i$ all have the same range (in this case $0 \le X_i \le 1$ for all $i$)"?
If you add up $n$ numbers, each in the interval $[0,1]$, then you get a number in the range $[0,n]$.
There is no assumption of units (amount of fuel, number of passengers, etc.) here, but implicitly, writing down $X_1 + \cdots + X_n$ implies that  for whatever physical quantity $X_i$ is supposed to model, the sum should make sense. Moreover the physical quantity should follow the probablistic assumption (uniform distribution): number of passengers does not make sense though, since presumably the number of passengers is a nonnegative integer, while $X_i$ takes on any value between $0$ and $1$.
A: It makes complete sense. 
Ask yourself this: What is the smallest possible value of $Z$? It is $0$.
What is the largest possible value? This occurs if all $X_i$ happen to be $1$, hence the largest value of $Z$ is $n$. So you can deduce that the range of $Z$ is $[0,n]$.
You stated

This result only makes sense to me if we assume that the $X_i$ actually all have the same range

They do. You told us each $X_i$ follows a unif$[0,1]$.
For "large" $n$, the sum of iid random variables converges to a normal distribution by the CLT.
A: If your random variables have different units (e.g. number of passengers, mileage, amount of fuel in the tank), then the addition (+) is not defined, and $X_1+X_2$ makes no sense.
To illustrate why addition of two uniform variables is not uniform, lets have a look at the sum of two dices
Dice #1:
$$p(d_1=1)=\frac{1}{6}$$
$$p(d_1=2)=\frac{1}{6}$$
$$p(d_1=3)=\frac{1}{6}$$
$$p(d_1=4)=\frac{1}{6}$$
$$p(d_1=5)=\frac{1}{6}$$
$$p(d_1=6)=\frac{1}{6}$$
Same for Dice #2
But, have a look at the sum
$$p(d_1+d_2=2)=\frac{1}{36}$$
$$p(d_1+d_2=3)=\frac{2}{36}$$
Already we see that the resulting distribution is not uniform
