Rescaling an exponential decay time to milliseconds

Question

A particle's decay time T is exponential with a mean of 5 seconds. You want to report it in milliseconds, so you define S = 1000\,T. What are the mean and variance of S (in milliseconds and milliseconds-squared)?

Accepted Answer

First get the moments of T in seconds. An exponential has variance equal to the mean squared, so \mathbb{E}[T] = 5 and \operatorname{Var}(T) = 5^2 = 25. Now apply the scaling rules for S = aT with a = 1000: - Mean scales linearly: \mathbb{E}[S] = 1000 	imes 5 = 5000 ms. - Variance scales by the square of the constant: \operatorname{Var}(S) = 1000^2 	imes 25 = 25{,}000{,}000 ms-squared. The standard deviation scales linearly by 1000, from 5 s to 5000 ms, consistent with \sqrt{25{,}000{,}000} = 5000. Takeaway. Rescaling by a multiplies the mean by a and the variance by a^2; a mean-5s exponent…

Rescaling an exponential decay time to milliseconds

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