I'm working on a homework problem and I'm not totally sure what the question is asking... The question reads:
"Consider the linear programming problem:
maximize $Ax_1+Bx_2$
subject to $x_1+x_2\leq 1$, $x_1\geq 0$, $x_2 \geq 0$
where the random coefficients $A\sim U(0,1)$ and $B\sim U(0,2)$ are independent random variables. Find the joint pmf of $X_1$ and $X_2$, the values of $x_1$ and $x_2$ that solve the linear program."
What does the maximize mean in this situation? What is meant by the values that "solve the linear program"? I'm not looking for help solving the actual problem, I just need a push in the right direction so I'm sure of the parameters for the pmf!
Temporarily imagine working with (positive) numbers $a$ and $b$. We want to maximize $ax_1+bx_2$ in the part of the first quadrant that is on or below the lin $x_1+x_2=1$.
It is obvious that the maximum is reached on the line $x_1+x_2=1$, and indeed reached (perhaps among other places) at a corner, that is, at $(1,0)$ or $(0,1)$. So the maximum value is $a$ or $b$, whichever is bigger. (If we have $a=b$ the maximum is reached everywhere on the line segment that joins $(1,0)$ to $(0,1)$.)
Now translate this to the random variable version.