Is there a name for this average?

Question

Let

$\lambda = \frac{\sum\limits_{n=1}^{N} \lambda_n B_n e^{-\lambda_n t_0}}{\sum\limits_{n=1}^{N} B_n e^{-\lambda_n t_0}}\,,$

with $\lambda_n$, $B_n$, $t_0$ real numbers.

I interpret $\lambda$ as some sort of average. Is there a name for it?

Answer 1

Generally,

$$ \lambda=\frac{\sum_nw_n\lambda_n}{\sum_nw_n} $$

is called a weighted average of the $\lambda_n$ with weights $w_n$. In your case, the weights $w_n=B_n\mathrm e^{-\lambda_nt_0}$ depend on the quantities $\lambda_n$ being averaged, which isn't usually the case, but one might still call it a weighted average in a wider sense.

Specifically,

$$ E=\frac{\sum_nB_nE_n\mathrm e^{-\beta E_n}}{\sum_nB_n\mathrm e^{-\beta E_n}} $$

is the mean energy in a system described by the Boltzmann distribution of statistical mechanics, where $E_n$ is the energy of state $n$, $B_n$ is its multiplicity, and $\beta=\frac1{kT}$ is the inverse temperature. An average taken with respect to the Boltzmann distribution is sometimes called a Boltzmann average.