In this subsection, $n$ is a natural integer greater than or equal to 2 and $X$ is a random variable taking values in $\mathbb{N}$, defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and following the uniform distribution on $\llbracket 0, n-1 \rrbracket$: $$\mathbb{P}(X = k) = \frac{1}{n} \text{ if } k \in \llbracket 0, n-1 \rrbracket \text{ and } \mathbb{P}(X = k) = 0 \text{ otherwise}$$ We assume in this question that $n$ is not prime: there exist integers $a$ and $b$, greater than or equal to 2, such that $n = ab$. a) Show that there exists a unique pair of integer-valued random variables $(Q, R)$ defined on $\Omega$ such that $$X = aQ + R \quad \text{and} \quad \forall \omega \in \Omega, R(\omega) \in \llbracket 0, a-1 \rrbracket$$ One may consider a Euclidean division. b) Specify the distribution of $(Q, R)$, then the distributions of $Q$ and $R$. c) Show that $X$ is decomposable. Deduce an expression of $G_{X}$ as a product of two non-constant polynomials that one will specify.
In this subsection, $n$ is a natural integer greater than or equal to 2 and $X$ is a random variable taking values in $\mathbb{N}$, defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and following the uniform distribution on $\llbracket 0, n-1 \rrbracket$:
$$\mathbb{P}(X = k) = \frac{1}{n} \text{ if } k \in \llbracket 0, n-1 \rrbracket \text{ and } \mathbb{P}(X = k) = 0 \text{ otherwise}$$
We assume in this question that $n$ is not prime: there exist integers $a$ and $b$, greater than or equal to 2, such that $n = ab$.
a) Show that there exists a unique pair of integer-valued random variables $(Q, R)$ defined on $\Omega$ such that
$$X = aQ + R \quad \text{and} \quad \forall \omega \in \Omega, R(\omega) \in \llbracket 0, a-1 \rrbracket$$
One may consider a Euclidean division.
b) Specify the distribution of $(Q, R)$, then the distributions of $Q$ and $R$.
c) Show that $X$ is decomposable. Deduce an expression of $G_{X}$ as a product of two non-constant polynomials that one will specify.