Named probability distributions are often explicitly presented as having a specific number of parameters. For example, even though the Poisson distribution PMF equation $p_K (k) = \frac{\lambda^k}{k!e^\lambda}$ has two variables, $k$ and $\lambda$, only $\lambda$ is said to be a parameter. It seems that the observed value of the random variable listed in the LHS of the PMF equation does not count as a parameter, but all other variables do. In other notation, the LHS might be written as $f(k; \lambda, [$other parameters, there aren't any for Poisson$])$, where the semicolon is used instead of a comma to delimit the observed value variables and the parameter variables.
Is this divide really a matter of conceptual importance in Probability Theory, or is it merely a decision made arbitrarily? Presented with only the RHS, $\frac{\lambda^k}{k!e^\lambda}$, I would classify $k$ and $\lambda$ symmetrically as independent variables, and the geometry expressed would be a surface in $3$-space. If we wanted to visualize the geometry as a curve in $2$-space, we'd take a slice of the surface holding either $k$ or $\lambda$ constant. In software, we could add a slider for the user to manipulate whichever variable we chose to hold constant within our slice. Thus, both variables may vary, but the curve only displays one variation at a time.
On one hand, it seems to me that the parameters of a probability distribution are nothing more than the set of variables assigned to sliders. If I decide to display $\lambda$ on the horizontal axis of the plane and create a slider for $k$, then I have represented the same information as the usual Poisson distribution PMF. Have I successfully switched the parameter and non-parameter of the Poisson distribution PMF by doing so? Is it still a Poisson distribution, or is $\lambda$ being a parameter essential to the nature of the Poisson distribution?
On the other hand, the set of parameters of a named distribution is often stated even when it is algebra rather than geometry under immediate consideration. If I have the wrong notion of a parameter above, then what is it really that distinguishes a parameter from a non-parameter?
Best Answer
The probablity mass function $p_{\lambda}$ is probabilitistically interpreted as so: If a random variable $X$ has distribution $\text{Poisson}(\lambda)$ then $$ \Pr(X = k) = p_{\lambda}(k). $$
Swapping the roles of $\lambda$ and $k$ gives a statement which doesn't make sense: If a random variable $X$ has distribution $\text{Poisson}(k)$ then $$ \Pr(X = \lambda) = p_{k}(\lambda). $$
You can see that the roles of $\lambda$ and $k$ are rather different. $\lambda$ indexes different probability distributions. On the other hand $k$ is a dummy variable used in the probability mass function. It is conceptually important that these two are treated differently.