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Let $X \sim \operatorname{Bin}(n, p)$, and $Y \sim \operatorname{Poisson}\; (\lambda)$. Let $$ T=X_{1}+X_{2}+\cdots+X_{Y}, $$ with $X_{i} \text{'s i. i. d}.\; \operatorname{Bin}(n, p)\;($and independent to $Y),$ and $$ S=Y_{1}+Y_{2}+\cdots+Y_{X}, $$ with $Y_{i} \text{'s i. i. d. Poisson}\; (\lambda) \;($and independent to $X).$ Compare Expectations of $T$ and $S$ and Variances of $T$ and $S.$
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