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Consider a sequence of Bernoulli trials with the probability of success $p$. Suppose you started the game with a run of successes followed by the run of failures (note that you can learn that unlucky run is over if and only if it is followed by a success). Let the random variable $X$ be the number of successful trials and $Y$ be the number of unsuccessful ones (we count as a run any sequence of one or more identical outcomes). Find

(b) Mean lengths of both runs, i.e. $E(X)$ and $E(Y)$.

(c) The correlation function of $E(XY)$.

(d) The covariance $\operatorname{Cov}(X,Y)$.

I am confused as I know it is a geometric distribution, and get confused as to how to go about doing this :/

Thanks for any help.

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    The following seems to be clear: one should condition on the first trial being a success; part (c) is meaningless; the OP did not show anything about what they tried; this is (homework).2012-11-19

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