Difference Between Conditional Distribution and Distribution Parameters

Question

Is there any connection between the parameter of a probability distribution and a conditional distribution?

What I am wondering about is, whether or not there is some notational shorthand or implicit mathematical rule that is just not taught, when we, say, calculate the expectation of a normal distribution with parameters $$X \sim \mathcal{N} (\mu,\sigma)$$ versus a distribution $$ E[X| M = \mu,\Sigma = \sigma] $$, as we're not using the same formulas for both, but we're conditioning both probability distributions on a scalar value.

I understand that this might be a stupid question, but I cannot find any sources on this on the internet, when I tried to research it myself.

Answer 1

To add my own thoughts after having finished my first probability theory course;

The difference lies in whether or not the parameter is a constant or a random variable held at a constant value. In a sense it is purely semantical, if all we care about is the distribution when we hold the other random variable constant.

To illustrate this, imagine we have an experiment where we flip $n$ fair coins and count the number of heads, and we would like to know the expected number of heads. Denote the number of heads with H.

For $n$ as a constant, we have $ H \sim Binomial(n,1/2)$, such that $E[H] = \frac{n}{2}$.

Now, assume that the number of coins available to us follows a uniform distribution, such that $N \sim U(0,1,...,10)$. Let the number of heads in such a scenario be denoted $H_2$. If we were to hold N constant, such that $N = n$, we would see that $E[H_2|N=n] = E[H]$.

However, this wouldn't be the true expectation of $H_2$ any longer. And as such we now have $E[H_2] \neq E[H]$, simply because n is now determined by the outcome of a random variable and is longer constant.

This is especially important when studying covariance and correlation, where changing a constant to be the result of a random variable held constant can have dramatic effects on the variance.