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ISI 2020 | PCB Mathematics | Question: 9

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Let $X_{i} \sim\left(i . i . d\right.$.) Bernoulli $\left(\frac{\lambda}{n}\right), n \geq \lambda \geq 0$.

$Y_{i} \sim\left(\right. i. i. d.)$ Poisson $\left(\frac{\lambda}{n}\right),\left\{X_{i}\right\}$ and $\left\{Y_{i}\right\}$ are independent.

Let $\displaystyle{}\sum_{i=1}^{n^{2}} X_{i}=T_{n}$ and $\displaystyle{}\sum_{i=1}^{n^{2}} Y_{i}=S_{n}$ (say).

Find the limiting distribution of $\dfrac{T_{n}}{S_{n}}$ as $n \rightarrow \infty.$
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