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Method of Indicators for computing expectation

  1. Sep 28, 2010 #1

    I have the following problem: Suppose you have a coin that has chance p of landing heads. Suppose you flip the coin n times and let X denote the number of 'head runs' in n flips. A 'head run' is defined as any sequence of heads. For example the sequence HHTHHHHHTTTTHHTHT contains 4 head runs. Given this information, compute E[X] and Var[X].

    I cannot understand how to set up the indicator method that will allow me to solve this problem quickly.
  2. jcsd
  3. Sep 29, 2010 #2
    Define [tex]X_i = 1[/tex] if flip i is the start of a run of heads,
    [tex]= 0[/tex] otherwise.

    To find E[X] you will need to compute [tex]E[\sum X_i][/tex].

    To find Var[X] you will need to compute, in addition, [tex]E[\sum X_i X_j][/tex], where the sum runs over all pairs i, j with i < j.
  4. Sep 30, 2010 #3
    Seems interesting, but hard problem. I suspect that the indicator method does not work in case of p [tex]\neq[/tex] 1/2. It is easier to consider a fair coin with p=1/2 at the beginning.
  5. Oct 1, 2010 #4
    After a messy, lengthy calculations (not the indicator method) an unexpectedly simple formula for the E(x) occurred:

    E(x)=p*(p+n*q) ; q=1-p

    I was shocked!! Indeed, there must be a simple probabilistic approach that replaces involved calculations and hard analysis. Maybe really the indicator method. Still needs to think about this problem.
  6. Oct 1, 2010 #5
    E(X1)=p and E(Xi)=pq for i>1 so E(X) = p+(n-1)pq = p^2 + npq
  7. Oct 2, 2010 #6
    Nice one bpet!!!

    I would not come into this as soon. Do you have a clue how to determine E(Xi*Xj) now, to calculate Var(X)?
  8. Oct 2, 2010 #7
    Thanks! Similar way, more cases to consider e.g. j=i, j=i+1, j>i+1.
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