Let two natural integers $A$ and $n$ such that $n \leqslant A$ and $p$ a real number between 0 and 1. We assume $pA \in \mathbb{N}$ and we denote $q = 1-p$. Let $X$ be a random variable such that $X \hookrightarrow \mathcal{H}(n, p, A)$, i.e. $$\mathbb{P}(X = k) = \frac{\binom{pA}{k}\binom{qA}{n-k}}{\binom{A}{n}} \quad \text{for all } k \in \llbracket 0, n \rrbracket.$$ Show that the sequence $(\mathbb{P}(X = k))_{k \in \mathbb{N}}$ is hypergeometric. Deduce an expression of the generating function of $X$ using a hypergeometric function.
Let two natural integers $A$ and $n$ such that $n \leqslant A$ and $p$ a real number between 0 and 1. We assume $pA \in \mathbb{N}$ and we denote $q = 1-p$. Let $X$ be a random variable such that $X \hookrightarrow \mathcal{H}(n, p, A)$, i.e.
$$\mathbb{P}(X = k) = \frac{\binom{pA}{k}\binom{qA}{n-k}}{\binom{A}{n}} \quad \text{for all } k \in \llbracket 0, n \rrbracket.$$
Show that the sequence $(\mathbb{P}(X = k))_{k \in \mathbb{N}}$ is hypergeometric. Deduce an expression of the generating function of $X$ using a hypergeometric function.