Is a data size in a binomial distribution random variable? Supposing that there is a binomial distribution ${\rm Bin}(m|N, \mu)$, I think usually $N$ and $\mu$ are parameters and not random variables (or events), thus the notation here ${\rm Bin}(m|N, \mu)$ seems somewhat misleading.
Here, the binomial distribution ${\rm Bin}(m|N, \mu)$ is
\begin{align*}
{\rm Bin}(m|N, \mu) = \begin{pmatrix} N \\ m\end{pmatrix} \mu^m (1 - \mu)^{N-m}
\end{align*}
in the PRML notation.
My Question:
However, is this notation really misleading? If not, what is the definition of $p(N)$ here?
If we would like to consider the Bayesian inference, we often introduce a beta distributions ${\rm Beta}(\mu | a, b)$ as the conjugate prior distribution of the parameter $\mu$.
Then, the PRML eq. (2.17) says that the posterior distribution of $p$ is given by
\begin{align*}
 p(\mu|m, N, a, b) \propto {\rm Bin}(m|N, \mu) {\rm Beta}(\mu | a, b)
\end{align*}
via the Bayes' theorem and I would like to show it explicitly. The $\propto$ means that if we ignore all factors independent of $\mu$ we obtain this expression.
To show this, at first I assumed $m$ and $a, b$ are conditionally independent given $\mu$ and $N$.Thus
\begin{align*}
p(m|N, \mu) = p(m|N, \mu, a, b) &=  \frac{p(m, N, \mu, a, b)}{p(N, \mu, a, b)} = \frac{p(m, N|\mu, a, b) p(\mu | a, b) p(a, b)}{p(N|\mu) p(\mu | a, b)} \\
&= \frac{p(m, N|\mu, a, b)p(a, b)}{p(N|\mu)}.
\end{align*}
Then we get
\begin{align*}
 p(m|N, \mu) p(\mu | a, b) &= \frac{p(m, N|\mu, a, b)p(a, b)}{p(N | \mu)} \frac{p(\mu, a, b)}{p(a, b)} \\
&= \frac{p(\mu, m, N, a, b)}{p(N|\mu)}\\
&= \frac{p(\mu|m, N, a, b)p(m, N, a, b)}{p(N|\mu)}.
\end{align*}
Obviously, the factor $p(N|\mu)^{-1}$ appears and contradicts the above statement.
Therefore I believe that $N$ (and hyperparameters $a, b$) are parameters, not random variables and thus we don't have to consider probabilities like $p(N)$ or $p(N|\mu)$.
Am I right in my thinking? If so, it would be even more appreciated if you could show the correct normalized expression with parameters separated.

Binomial distribution has two parameters: sample size and probability of success. The parameters can either be fixed or be random variables, can be known or need to be estimated. If they are to be estimated, then depending on your preferred approach they can be treated as random variables (Bayesian school of thought) or not (frequentist or likelihoodist). So both cases are possible and you need to consult the book about what exactly the author says.
That said, using conditional “$| $” (vs “$; $”) notation often, but not necessary, suggests that the author treats them as random variables. If they talk about posterior, that this is clearly the case, since there are no non-Bayesian posteriors.
