Independent decay times are drawn from the exponential density ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

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Problem

Independent decay times

t_1,\dots,t_N

are drawn from the exponential density

f(t) = \tfrac{1}{\tau}e^{-t/\tau}

t\ge 0

, with

\tau = 5.0\,\mu\mathrm{s}

, and

N = 25

decays are recorded with no truncation of the time window. Write

u

for the transform variable, to keep it apart from the decay times. ((a)) Write the characteristic function

\phi(u)

of a single decay time, and use the scaling rule to obtain the characteristic function

\phi_{\bar t}(u)

of the sample mean

\bar t = \frac1N\sum_{i=1}^{N} t_i

. ((b)) Expand

\ln\phi_{\bar t}(u)

to second order in

u

and read off

\langle\bar t\rangle

and

\mathrm{var}(\bar t)

. Is

\bar t

unbiased? ((c)) Find the limit of

\phi_{\bar t}(u)

N\to\infty

at fixed

u

, and say what it means for the distribution of

\bar t

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