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).
(a) The joint probability P(X = n; Y = m)
(b) Mean lengths of both runs, i.e. E(X) and E(Y)
(c) The correlation function of E(XY)
(d) The covariance Cov(XY)
This was a past exam question and I have a mid-term coming up and I have a few questions :
Is this a geometric distribution?
For a) is the joint probability = q^(n+m)p^2
Basically I'm a bit confused on how to do part b and c.
Any Help would be appreciated. Thanks for your time.
Re: Probability Help
In terms of the run problem, you will know that you get a specific set of values corresponding to the run which are n X's followed by m Y's.
Since every trial is independent with a given probability of obtaining an individual X with some probability p and an individual Y with an individual probability 1-p, then you can use the independence criterion P(A and B) = P(A)P(B) to get the final probability.
For the rest, you should consider how to calculate the distribution symbolically (using the above) to get means and covariance terms with those formulae.