grandes-ecoles 2019 Q10

grandes-ecoles · France · centrale-maths1__psi Probability Generating Functions Deriving moments or distribution from a PGF
We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:
  • we randomly draw a ball from the urn;
  • we replace the drawn ball in the urn;
  • we add to the urn a ball of the same color as the drawn ball.
We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws. We denote by $g_{n}$ the generating function of the random variable $X_{n}$.
Prove that, for all integers $n \in \mathbb{N}^{*}$ and all real $t$, $$g_{n}(t) = \frac{1}{n+1} \sum_{k=1}^{n+1} t^{k}$$ 
We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:
\begin{itemize}
  \item we randomly draw a ball from the urn;
  \item we replace the drawn ball in the urn;
  \item we add to the urn a ball of the same color as the drawn ball.
\end{itemize}
We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws. We denote by $g_{n}$ the generating function of the random variable $X_{n}$.

Prove that, for all integers $n \in \mathbb{N}^{*}$ and all real $t$,
$$g_{n}(t) = \frac{1}{n+1} \sum_{k=1}^{n+1} t^{k}$$
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