Jane Street Interview Question: If I flip 100 coins and then ... | Glassdoor.co.uk

Interview Question

Quantitative Trader Interview London, England

If I flip 100 coins and then multiply the number of heads

  by the number of tails, what is the expected value of that number? Can you give a confidence interval on this number?

Interview Answer

3 Answers


I would estimate fairly close to 2500. The payoff function j*(100-j) is quadratic and the density is approximately Gaussian with mean 50. Both of these are highly concentrated around their modes (both of which are 50).

The actual answer turns out to be 2475.

Jacob on 24 Sep 2020

Y is a binomial distribution. Hence it has mean n*p and variance n*p*(1-p). We seek

E[Y*(n-Y)] = n*E[Y] - E[Y^2] = n*(n*p) - (Var[Y] + E[Y]^2) = n^2 * p - n*p*(1-p) - (n*p)^2 = n*(n-1)*p*(1-p)

In the case of fair coin (p=0.5) and n=100, we have:

E[Y*(100-Y)] = 100*99*0.5*0.5 = 2475.

Anonymous on 5 Oct 2020

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