grandes-ecoles 2019 Q11

grandes-ecoles · France · centrale-maths1__psi Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process
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.
Identify the distribution of $X_{n}$ and give its expectation. 
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.

Identify the distribution of $X_{n}$ and give its expectation.
Paper Questions
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