$$X = \sum_i^n x_i$$
,where $x_i \in (a_i,b_i)$ are independent uniform random variables, how does one find the probability distribution of $X$.
The sum of $n$ iid random variables with (continuous) uniform distribution on $[0,1]$ has distribution called the Irwin-Hall distribution. Some details about the distribution, including the cdf, can be found at the above link. One can then get corresponding information for uniforms on $]a,b]$ by linear transformation.