grandes-ecoles 2019 Q7

grandes-ecoles · France · centrale-maths1__psi Probability Generating Functions Explicit computation of a PGF or characteristic function

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 $X_{n}$, where $g_{n}(t) = \sum_{k=0}^{+\infty} P(X_{n} = k) t^{k}$.
Determine the distributions of $X_{1}, X_{2}$ and $X_{3}$ and then the functions $g_{1}, g_{2}$ and $g_{3}$.

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 $X_{n}$, where $g_{n}(t) = \sum_{k=0}^{+\infty} P(X_{n} = k) t^{k}$.

Determine the distributions of $X_{1}, X_{2}$ and $X_{3}$ and then the functions $g_{1}, g_{2}$ and $g_{3}$.

Paper Questions

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