Let $n$ be a non-zero natural integer. For $i \in \llbracket 1 , n \rrbracket$, $X _ { i }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following a Bernoulli distribution with parameter $\lambda / n$. We assume that $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are mutually independent and we set $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$.
For $t \in \mathbb { R }$, calculate $\lim _ { n \rightarrow + \infty } M _ { S _ { n } } ( t )$.

Let $n$ be a non-zero natural integer. For $i \in \llbracket 1 , n \rrbracket$, $X _ { i }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following a Bernoulli distribution with parameter $\lambda / n$. We assume that $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are mutually independent and we set $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$.
Compare with the results of question 8.

Let $X$ be a random variable following the distribution $\mathcal{H}(n, p, A)$, i.e. $$\mathbb{P}(X = k) = \frac{\binom{pA}{k}\binom{qA}{n-k}}{\binom{A}{n}} \quad \text{for all } k \in \llbracket 0, n \rrbracket.$$ We fix $n$ and $p$. Let $k \in \llbracket 0, n \rrbracket$. Show that $$\lim_{A \to +\infty} \mathbb{P}(X = k) = \binom{n}{k} p^k (1-p)^{n-k}.$$

Let $X$ be a random variable following the distribution $\mathcal{H}(n, p, A)$. We fix $n$ and $p$. We have shown that $$\lim_{A \to +\infty} \mathbb{P}(X = k) = \binom{n}{k} p^k (1-p)^{n-k}.$$ Interpret this result in connection with those obtained for the expectation and variance of $Y$.

Let $\alpha$ be a strictly positive real number. For all natural number $n$ such that $n > \lfloor \alpha \rfloor$, we denote by $B_n$ a binomial random variable with parameters $n$ and $\frac{\alpha}{n}$. For all natural number $k$, determine $$\lim_{n \rightarrow +\infty} P\left(B_n = k\right)$$ One may use the previous question.