Paper 2, Section II, F

Probability | Part IA, 2011

(a) State Markov's inequality.

(b) Let rr be a given positive integer. You toss an unbiased coin repeatedly until the first head appears, which occurs on the H1H_{1} th toss. Next, I toss the same coin until I get my first tail, which occurs on my T1T_{1} th toss. Then you continue until you get your second head with a further H2H_{2} tosses; then I continue with a further T2T_{2} tosses until my second tail. We continue for rr turns like this, and generate a sequence H1,T1,H2,T2,,HrH_{1}, T_{1}, H_{2}, T_{2}, \ldots, H_{r}, TrT_{r} of random variables. The total number of tosses made is YrY_{r}. (For example, for r=2r=2, a sequence of outcomes tthtttthhhtt t h|t| t t t h \mid h h t gives H1=3,T1=1,H2=4,T2=3H_{1}=3, T_{1}=1, H_{2}=4, T_{2}=3 and Y2=11Y_{2}=11.)

Find the probability-generating functions of the random variables HjH_{j} and TjT_{j}. Hence or otherwise obtain the mean values EHj\mathbb{E} H_{j} and ETj\mathbb{E} T_{j}.

Obtain the probability-generating function of the random variable YrY_{r}, and find the mean value EYr\mathbb{E} Y_{r}.

Prove that, for n2rn \geqslant 2 r,

P(Yr=n)=12n(n12r1)\mathbb{P}\left(Y_{r}=n\right)=\frac{1}{2^{n}}\left(\begin{array}{c} n-1 \\ 2 r-1 \end{array}\right)

For r=1r=1, calculate P(Y15)\mathbb{P}\left(Y_{1} \geqslant 5\right), and confirm that it satisfies Markov's inequality.

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