Nneetquant
Get started

Draws to pass two

HardProbabilityRandom VariablesExpected Value~11m

Draw X1,X2,…X_1, X_2, \dots independently from Uniform(0,1)\text{Uniform}(0,1) and let N=min⁡{ n:X1+⋯+Xn>2 }N = \min\{\,n : X_1 + \dots + X_n > 2\,\}. Find E[N]\mathbb{E}[N] (give a numeric value to four decimal places).

Show hints (2)+
  1. E[N]=∑n≥0P(Sn≤2)\mathbb{E}[N]=\sum_{n\ge0}\mathbb{P}(S_n\le 2). For 1≤x≤21\le x\le2 the simplex volume needs a correction: xn/n!−n(x−1)n/n!x^n/n! - n(x-1)^n/n!.
  2. At x=2x=2 this sums to e2−ee^2 - e. (The naive exe^x formula only holds for x≤1x\le1.)

Answer

Reveal answer →

4.6708 (± 0.001)

Want the full step-by-step worked solution? It's part of Premium - along with a worked solution for every question in the bank.

Practise freeUnlock solutions

Asked at: Jane Street, Two Sigma

Related questions