At any time, a dog has the probability of p to bark. What's the probability that this dog did not bark in the past T seconds?
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This is phrased as a continuous time question: the probability $p$ isn't referring to barking within a minute, or second, or microsecond, but at any time. That indicates continuous time. The distribution that describes the probability of an event that occurs at a constant rate is the exponential distribution. Having a probability $p$ of barking at a "moment" -- an infinitesimal unit of time, means that: $p = \lim_{t \rightarrow 0} P($Bark at time $<t)/t = \lim_{t \rightarrow 0} F(t)/t = f(0) = \lambda e^{-\lambda 0} = \lambda$ For the exponential. So the rate parameter is $p$. The question asks for the probability of not barking in an interval of time $T$. That is given by $1 - F(T) = 1-(1-e^{-pT}) = e^{-pT}$. That makes the answer $e^{-pT}$. |
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Hint: What are the odds that an event happens twice in a row? three times? in relation to it happening once? Also, what are the odds of an event not happening, as opposed to it happening? |
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It should be (1-P)^T.Because 1-P is the probability that the dog has not barked in the last 1 sec (assuming that p is the probability that it does not bark in a given second). |
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This should have Poisson distribution. http://en.wikipedia.org/wiki/Poisson_distribution |
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