Problem 2

How many time do we need to toss to estimate the fairness of a coin?

Checking whether a coin is fair


Problem 1

The expectation times of sampling to get a number twice from a discrete uniform distribution .

denotes the probability that are distinct numbers, and .

Solution

Expectation:

where

Variance:

I don’t find a good way to calculate it yet.

pr[x_,n_] := x * Product[(n-i), {i, 0, x - 1}] / n^(x+1)

en1[n_] := Sum[i * pr[i, n], {i, 1, n}]

ens1[n_] := Sum[i * i * pr[i, n], {i, 1, n}]


prlist[n_] := NestList[{#[[1]] + 1, (#[[1]] + 1) / n * (n - #[[1]]) / #[[1]] * #[[2]]} &, {1, 1/n}, n - 1]

en2[n_] := Total[Apply[Times, prlist[n], {1}]]

ens2[n_] := Total[Apply[Function[{a,b}, a*a*b], prlist[n], {1}]]


en[n_] := (E/n)^n * Gamma[n+1,n] - 1

varn[n_] := ens2[n] - en[n]^2

N[en[10], 10]

N[en[100], 10]

N[en[200], 10]

N[varn[10], 10]

N[varn[100], 10]

N[varn[200], 10]

cdf[n_,p_] := 
  Catch[
    Fold[
      Function[{s, x}, Print[{x[[1]],N[x[[2]] + s]}];
        If[x[[2]] + s >= p, 
           Throw[x[[1]]], 
           s + x[[2]]]], 
      0, prlist[n]]]
           
cdf[1000, 0.99]

Problem 0

Sample numbers without replacement from . Let . Find and .

Solution

en := m/(n+m) * Sum[i, {i,1,n+m}]

ens := m/(n+m) * Sum[i^2, {i, 1, n+m}] + m(m-1)/((m+n)(n+m-1)) * Sum[Sum[2*i*j, {j, i+1, n+m}], {i,1,n+m-1}]

varn := Simplify[ens - en^2]