= Log-linear exponential-decay fit
{title2=$y=\ln(R/R_*),\quad m=-1/\tau,\quad c=\ln(R_0/R_*)$}
An <exponential decay> rate $R=R_0e^{-t/\tau}$ becomes a straight line in $x=t$ and $y=\ln(R/R_*)$, where $R_*$ is a fixed reference rate. Constant small fractional rate errors become approximately constant vertical errors $s\simeq\sigma_R/R$ by <propagation of uncertainty>. With independent Gaussian log-rate errors, <linear least squares> gives $\widehat m=S_{xy}/S_{xx}$ and $\widehat c=\bar y-\widehat m\bar x$.
The <covariance matrix> is obtained by applying the variance rules to these linear estimators:
$$
\operatorname{Var}\widehat m=\frac{s^2}{S_{xx}},\quad
\operatorname{Var}\widehat c=s^2\left(\frac1n+\frac{\bar x^2}{S_{xx}}\right),\quad
\operatorname{Cov}(\widehat m,\widehat c)=-\frac{s^2\bar x}{S_{xx}}.
$$
Transforming back gives $\widehat\tau=-1/\widehat m$, $\sigma_\tau\simeq\sigma_m/\widehat m^2$, $\widehat R_0=R_*e^{\widehat c}$ and $\sigma_{R_0}\simeq\widehat R_0\sigma_c$. If a candidate lifetime fixes the slope $m_a$, the intercept must be refitted as $c_a=\bar y-m_a\bar x$. The profiled objective obeys
$$
\chi^2(m_a,c_a)=\chi^2_{\min}+\frac{S_{xx}}{s^2}(m_a-\widehat m)^2.
$$
This follows by separating the centred slope residual from the unrestricted <regression residual>, whose inner product with the design columns is zero by the <normal equations for linear least squares>. Known supplied measurement errors are not rescaled by the observed reduced chi-squared. This log-error model is an approximation for proportional errors, not an exact replacement for every raw-count likelihood.
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