Notation of marginal probabilities

Question

I found a (to me) strange notation concerning marginal probabilities I don’t understand. Unfortunately I will include picture of the notation.

Does it mean x=2.5, shouldn’t it be 0 then? Do they mean the cumulative distribution fuchtion?

Thanks

Answer 1

They mean the density, which is equal to the derivative of the CDF wherever this one is differentiable.

The notation for a CDF would be $F_X(x)$ (with a capital $F$).

Both are linked by the fundamental $$F_X(x)=\int_{-\infty}^x f_X(t)dt$$

Note that such a density does not necessarily exist. When it does, $X$ is called absolutely continuous.

Note also that $f_X$ has little to do with the probability mass function (as you seem to believe when you say that it has to be $0$), which is denoted by $p_X$.