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## Q. 4.26

A radioactive mass emits particles according to a Poisson process at a mean rate of 15 particles per minute. At some point, a clock is started. What is the probability that more than 5 seconds will elapse before the next emission? What is the mean waiting time until the next particle is emitted?

## Verified Solution

We will measure time in seconds. Let T denote the time in seconds that elapses before the next particle is emitted. The mean rate of emissions is 0.25 per second, so the rate parameter is λ = 0.25, and T ∼ Exp(0.25). The probability that more than 5 seconds will elapse before the next emission is equal to

P(T > 5) = 1 – P(T ≤ 5)

= 1 – (1 – $e^{-0.25(5)}$)
= $e^{-1.25}$
= 0.2865

The mean waiting time is $μ_{T} = \frac{1}{0.25}$ = 4 seconds.