grandes-ecoles 2019 Q9

grandes-ecoles · France · centrale-maths1__psi Probability Generating Functions Recursive or recurrence relation via PGF coefficients
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 $X_{n}$.
Using the recurrence relation from question 8, deduce that, for all integers $n$ greater than or equal to 1 and all real $t$, $$g_{n}(t) = \frac{t^{2} - t}{n+1} g_{n-1}^{\prime}(t) + g_{n-1}(t)$$ 
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 $X_{n}$.

Using the recurrence relation from question 8, deduce that, for all integers $n$ greater than or equal to 1 and all real $t$,
$$g_{n}(t) = \frac{t^{2} - t}{n+1} g_{n-1}^{\prime}(t) + g_{n-1}(t)$$
Paper Questions
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25 Q26 Q27 Q28 Q29 Q30 Q31 Q32 Q33 Q34 Q35 Q36 Q37 Q38 Q39