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Suppose the counts recorded by a Geiger counter follow a Poisson process with an average of three counts per minute. (a) What is the probability that there are no counts in a 30-second interval? (b) What is the probability that the first count occurs in less than 10 seconds? (c) What is the probability that the first count occurs between 1 and 2 minutes after start-up? (d) What is the mean time between counts? (e) What is the standard deviation of the time between counts? (f) Determine x, such that the probability that at least one count occurs before time x minutes is 0.95.

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Given:

λ=3 per minute\lambda=3\text{ per minute}

Formula Poisson probability:

P(X=k)=λkeλk!P(X=k)=\dfrac{\lambda^k e^{-\lambda}}{k!}

Formula exponential probability density function and cumulative distribution function::

f(x)=λeλxf(x)=\lambda e^{-\lambda x}

F(x)=P(Xx)=1eλxF(x)=P(X\leq x)=1-e^{-\lambda x}

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