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M.g.f. help. Mean, Variance, and standard deviation.

  1. Feb 27, 2004 #1
    I am having trouble with this question.

    Let X equal the number of flips of a fair coin that are required to observe the same face on consecutive flips.
    (a) Find the p.m.f. of X.
    if found the p.m.f. to be [tex]f(x) = (\frac{1}{2})^{x-1}[/tex] for [tex]x=2,3,4,...[/tex]

    (b) Give the values of the mean, variance and standard deviation of X.
    For this one i found the m.g.f. to be [tex]M(t) = E(e^{tx}) = \sum_{x \in S} e^{tx}f(x)[/tex]
    [tex]M^{'}(0) = xf(x) = E(X)[/tex] which is the mean.
    Then
    [tex]M^{''}(0) = x^{2}f(x) = E(X)[/tex]
    [tex]Var(X) = M^{''}(0) - [M^{'}(0)]^2[/tex]

    Is this correct?

    Than after that i realized that 2 rolls of the dice was the minimum to get the same face on two consecutive flips. So i made [tex]S={1,2}[/tex] and evaluted them like this getting the Mean = 2 and Variance = 4 which is not correct.
    What am i doing wrong? Do i have to use Infinite series or something?

    I haven't even started on the standard deviation.

    (c) Find the values of
    [tex](i) P(X \leq 3)[/tex]
    [tex](ii) P(X \geq 5)[/tex]
    [tex](iii) P(X = 3)[/tex]

    I can't start on these until i get part (b)
     
  2. jcsd
  3. Mar 2, 2004 #2

    matt grime

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    Science Advisor
    Homework Helper

    S must be the set {2,3,4....}. yes you need to evalaute the infinite sums, but they aren't hard to do. the sum of nr^n and n(n-1)r^n are quite well known and you should be able to find them.

    you can do c without doing b; it doens't involve moments at all.


    probability it happens on the first throw is 0, probability on the second is 1/2, and on the third is 1/4 so the first probability is 3/4

    x greater than 5 is 1- prob of on the 2nd 3rd or 4th, which is 1-1/2-1/4-1/8

    on the fifth exactly
    well one of HTHTT or THTHH must happen and the probability is....?
     
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