I have an infinite number of known probability density functions $f_1(x),f_2(x),f_3(x),...$. The PDFs $f_k(x)=\sum_{j=1}^k v(A+j-1)e^{-v(A+j-1)x}\binom{k}{j-1}q^{j-1}(1-q)^{k-j-1}$.
Let $g_i(x)=f_1*f_2*\dots*f_i (x)$,
where $∗$ represents the convolution operation. Denote by $G_i$ and $F_i$ the CDFs corresponding to the PDFs $g_i$ and $f_i$ respectively.
Now, given $T\in\mathbb{R}_{>0}$, I have a discrete random variable $N_T$, where
$Prob(N_T=i)=G_i(T)-g_i*F_{i+1}(T)=G_i(T)-G_i*f_{i+1}(T)$,
again $∗$ represents the convolution operation.
Now I need to prove that the variance of $N_T$ increases with $T$.