The sum of cumulative distribution function.

Question

Let, $F_1$ and $F_2$ are cumulative distribution functions. Find $a,b \in \mathbb{R}$ such that, $aF_1+bF_2$ is "cdf".

Answer 1

The condition that $aF_1+bF_2$ be a cdf are 1) $\lim_{x\to\infty} (aF_1+bF_2)(x)=a+b$ be 1; 2) $\lim_{x\to-\infty} (aF_1+bF_2)(x)=0$, satisfied for any finite $a,b$; 3) $aF_1+bF_2$ lie in $[0,1]$ and 4) are nondecreasing. The last conditions will be satisfied for nonnegative $a,b$, and putting everything together, we want $a,b\in[0,1]$ such that $a+b=1$, a "convex combination."

Depending on the $F_1,F_2$, there might be other combinations that lead to another cdf. E.g., if $F_1=F_2$, you could just take any combination satisfying $a+b=1$, $a,b\in\mathbb{R}$.