Poisson distribution inequality for sum of rates

Question

Given $s,t \in \mathbb{R}^+$ and $i,j \in \mathbb{N}$, Let $X,Y,Z$ be random variables with Poisson distribution with rate $s,t$ and $s+t$, respectively. Is it true that $$ P(Z = i+j) \geq P(X=i) \cdot P(Y=j).$$ I would need to have $$ \frac{s^i}{i!}\cdot \frac{t^j}{j!} \leq \frac{(s+t)^{i+j}}{(i+j)!}.$$ By the Young's inequality with $p = (i+j)/i$ and $q = (i+j)/j$, for every $a,b \in \mathbb{R}^+$, $$ ab \leq \frac{a^{\frac{i+j}{i}}}{\frac{i+j}{i}} + \frac{b^{\frac{i+j}{j}}}{\frac{i+j}{j}}. $$ I've tried different values to $a,b$ but still I cannot do it. For example, for $a = (s(i+j)/i)^{i/(i+j)}$ and $b$ analogous, we have $$ \frac{s^i }{(\frac{i}{i+j})^i} \cdot \frac{t^j }{(\frac{j}{i+j})^j} \leq (s+t)^{i+j}$$ Unfortunately, from there I couldn't go further. Any suggestion?

Answer 1

I think I have an idea using coupling. I define $X \sim \text{Pois}(s)$,$Y \sim \text{Pois}(t)$ independently in $P$. Then $Z = X+Y \sim \text{Pois}(t+s)$. Then $$ P(X = i)\cdot P(Y=j) = P(X=i,Y=j) \leq P(X+Y = i+j) = P(Z = i+j).$$ Do you think is right?