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.