Hamilton's inclusive fitness approach
From the limited information, I can provide the following but I am not sure if this is what you are looking for. Also, I still don't see the statement where the author concludes we get the Linear regression model $E(q_i) = Aq_i + C$ which is odd notation since it says the expected value is a linear regression. In fact, if it is a linear regression, it should say $q_j = Aq_i + C$.
$$
A = \frac{\text{cov}(q_i,q_j)}{\text{var}(q_i)}\quad\text{and}\quad C = E[q_j] - \frac{\text{cov}(q_i,q_j)}{\text{var}(q_i)}E[q_i].
$$
Now, we can write the variance and covariance as
\begin{align}
\text{var}(q_i) &amp;= E[q_i^2]-E^2[q_i]\\
\text{cov}(q_i,q_j) &amp;= E[q_iq_j]-E[q_i]E[q_j]
\end{align}
where the expected value of a discrete random variable $X$ is calculated as
$$
E[X] = \sum_{i=1}^Nx_ip_X[x_i]
$$
where $p_X[x_i]$ is probability of $x_i$ and $N$ can be countable infinite.



The mean of conditional PDFs comes up in optimal prediction where the minimum mean square error is $E_{Y\mid X}[Y\mid x]$. This optimal prediction covers linear and nonlinear. For the standard Gaussian PDF, the optimal prediction will be linear since $E_{Y\mid X}[Y\mid x]=\rho x$ where $\rho$ is the correlation coefficient. I am going to short hand $E_{Y\mid X}[Y\mid x]$ to $E[Y\mid x]$
$$
E[Y\mid x] = \mu_Y + \frac{\rho\sigma_Y}{\sigma_X}(x-\mu_X)
$$
where $\mu_i$ $i=X,Y$ is the mean and $\sigma_i$ is the standard deviation. If $X$ and $Y$ are not Gaussian, then the model can be nonlinear. Are there examples of non Gaussian be linear probably.