I am not expert in probability. And it is a question to me.
Imagine $p_k$ are random numbers with uniform distribution over [0,1]. And each $r_k$ are definite real numbers that we know. I want to know the probability of:
$$\Sigma_{k=1}^N r_k p_k \geq \gamma$$
N is a fixed and finite natural number and gamma is a number that we have decided. Is there any function for giving such probability?
PS. I know it will tend to Normal when N tends to $\infty$ but I need the solution for finite $N$.
Let $\displaystyle Y=\Sigma_{k=1}^N r_k p_k$. Assuming independence, and recalling that the characteristic function of $\displaystyle p_k$ is $\displaystyle \phi_k(t)=\frac{e^{it}-1}{it}$, we may write $$\phi_Y(t)=\frac{\prod_{k=1}^N(e^{ir_kt}-1)}{\prod_{k=1}^Nr_k(it)^N}.$$
Now using Gil-Pelaez inversion recipe you can calculate probability of $\displaystyle Y \geq \gamma$.