I am having trouble with this question.(adsbygoogle = window.adsbygoogle || []).push({});

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)

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

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