Time until the fifth radioactive decay

Question

The time between successive radioactive decays is exponential with a mean of 3 minutes, and the gaps are independent. Let T be the time until the 5th decay. What are the mean and variance of T?

Accepted Answer

Waiting for the 5th decay means summing 5 independent exponential gaps, which is an Erlang. Each gap has mean 1/\lambda = 3 minutes and variance 1/\lambda^2 = 3^2 = 9. Means and variances of independent pieces add: \mathbb{E}[T] = 5 	imes 3 = 15 	ext{ min}, \qquad \operatorname{Var}(T) = 5 	imes 9 = 45. Standard deviation \sqrt{45} \approx 6.71 minutes. Takeaway. Summing 5 independent exponentials with mean 3 gives an Erlang with mean n/\lambda = 15 and variance n/\lambda^2 = 45.

Time until the fifth radioactive decay

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