Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section E / 27 / a

Assume all contributing decay rates satisfy ${\nu }_l>0$ and the summed expression is a nonnegative density. ThenThus ${\sum }_l{A}_l=1$ normalizes the first-passage distribution. A zero decay rate would require separate treatment of a nonabsorbing component; eventual absorption is implicit here. Individual spectral coefficients need not all be positive, but the full density must be.

For an unnormalized representation, divide the first moment by its total mass. Integration by parts givesConsequentlyFor normalized amplitudes the denominator is one. A mode with small weight can still dominate the mean if its lifetime $1/{\nu }_l$ is sufficiently large; this distinction matters when interpreting the later graph.