grandes-ecoles 2019 Q23

grandes-ecoles · France · centrale-maths2__mp Discrete Probability Distributions Proof of Distributional Properties or Symmetry

Let $n$ be a non-zero natural number and let $X_n$ be a random variable that follows a uniform distribution on $D_n$. Show that there exist random variables $V_1, \ldots, V_n$ mutually independent, each following a Bernoulli distribution with parameter $1/2$, and such that $$X_n = \sum_{k=1}^{n} \frac{V_k}{2^k}.$$

Let $n$ be a non-zero natural number and let $X_n$ be a random variable that follows a uniform distribution on $D_n$. Show that there exist random variables $V_1, \ldots, V_n$ mutually independent, each following a Bernoulli distribution with parameter $1/2$, and such that
$$X_n = \sum_{k=1}^{n} \frac{V_k}{2^k}.$$

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