Solution
= Solution
For an unnormalized representation, divide the first moment by its total mass. Integration by parts gives
$$
\int_0^\infty t\nu_le^{-\nu_lt}\,dt=\frac1{\nu_l}.
$$
Consequently
$$
\boxed{\mathcal T=\frac{\sum_lA_l/\nu_l}{\sum_lA_l}.}
$$
For 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.