The random variables $X$ and $Y$ take integer values $x$ and $y$, both $≥ 1$, and such that $2x + y ≤ 2a$, where a is an integer greater than $1$. The joint probability within this region is given by:
$$P(X=x,Y =y)=c(2x+y)$$
where c is a constant and it is zero elsewhere.
Show that the marginal probability $P (X = x)$ is given by:
$$P(X=x) = \frac{6(a−x)(2x+2a+1)}{a(a−1)(8a+5)}$$
How do I calculate the value of $c$?
Basically you don’t need to use the integrals to solve this problem, noting that x is discrete.
$P(x=k)=\sum_{t=1}^{2a-2k}{P(x=k, y=t)} $
Which will lead to the final answer.