We are given a finite collection of random variables, $X_1,\ldots,X_n$. Each can take exactly $k$ different real values, with the same probability, $1/k$, but a priori these values are not the same for each of the variables. Thus, if $X_i$ can assume values $x_1^i,\ldots,x_k^i$, its probability distribution is given by: $$\mathbb{P}[X_i=x]=\begin{cases}1/k\quad if\ x\in {x_1^i,\ldots,x_k^i}\\ 0\quad otherwise \end{cases}$$ We now take a linear combination of the random variables, and define $$ Y=\sum_i a_iX_i,\quad a_i>0, s.t. \sum_i a_i =1$$ How can we describe the probability distribution of this new random variable? My final goal would be, given $k\in\mathbb{R}$, determining the right coefficients to maximise $$\mathbb{P}[Y>k]$$
What I would like to do is to write the new probability distribution of the the random variable $Y$, that of course will be dependant on the coefficients $\left\{a_i\right\}_i$, and if possible its associated cumulative distribution. So far I haven't been able to get to any "closed form" for the probability distribution of "Y". My hunch would be to say that "Y" still has an equally distributed probability function, in the generic case, and that I should therefore choose the coefficients $a_i$ in such a way that its support is "as shifted as possible" so that it would be greater than $k$ as much as possible. Am I on the right path? Any help would be very appreciated. Thank you!