Geometric Random Variable ProblemA fair coin is tossed repeatedly and independently until two consecutive heads or two consecutive tails appear. Find the PMF, the expected value, and the variance of the number of tosses.

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yermarvg · Accepted Answer

Step 1Let our random variable, the count of tosses until termination, be denoted as N.If the termination is the first occurance of two consecutive tosses of the same result, then you must toss a pattern like …THTHTHTT or …HTHTHTHH . Ie: Always alternating until the very end. Obviously.Step 2So, to stop on toss # 2k, for any   k  ∈  {  1  ,  2  ,  …  }, you must toss either:   k  −  1 consecutive TH pairs then a TT pair, or   k  −  1 consecutive HT pairs then a HH pair. So what is       P    (  N  =  2  k  )?Step 2Likewise to stop on toss #   2  k  +  1, for any   k  ∈  {  1  ,  2  ,  …  }, you must toss either: k consecutive TH pairs then a single H, or k consecutive HT pairs then a single T. So what is       P    (  N  =  2  k  +  1  )?Then you have a piecewise function for your probability depending on whether the count you seek is even or odd.      P    (  N      =    n  )     =         {                                                                    _                                                :                          n          ∈          2                                    N                        +                                                                                        _                                                :                          n          ∈          2                                    N                        +                                +                    1

A fair coin is tossed repeatedly and independently until two consecutive heads or two consecutive tails appear. Find the PMF, the expected value, and the variance of the number of tosses.

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