Two independent waiting times and are exponentially distributed

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

Error, Chance and the Machine, Ch Estimators and the information in data

Problem

Two independent waiting times

X_1

and

X_2

are exponentially distributed,

f(x) = \lambda e^{-\lambda x}

for

x \ge 0

, with rates

\lambda_1 = 2\,\mu\mathrm{s}^{-1}

and

\lambda_2 = 4\,\mu\mathrm{s}^{-1}

. The characteristic function of a single exponential variable of rate

\lambda

\phi(t) = \lambda/(\lambda - it)

. Let

S = X_1 + X_2

. ((a)) Write the characteristic function

\phi_S(t)

S

. ((b)) Expand

\ln\phi_S(t)

to second order in

t

and read off

\langle S\rangle

and

\mathrm{var}(S)

. ((c)) Check these against the direct sum of the means and variances of

X_1

and

X_2

separately.

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