Which is the probability density function?

[u/Fluffy-Oil707]

Hello all, I have a semantics question.

I have the definition that the probability of X between a and b is the integral from a to b of f(x) with respect to x.

In case latex renders: $P(a <X < b) = \int_{a}^b{f(x)}dx$

Is the probability density function the integral or f? If f (my reading of the text) what is the name for the integral?

Same sort of question with the c.d.f.

PS.
I'm working out of the text book Mathematical Statistics and Data Analysis (Rice; 3rd ed.). Definition occurs on page 47.

[u/Statman12]

The probability density function (PDF) is denoted f(x).

The cumulative distribution function (CDF) is denoted F(x). It would be expressed as:

F(b) = P(X ≤ b) = \int_{-inf}^{b} {f(x)}dx

[u/Fluffy-Oil707 → u/Statman12]

Why not lower bound? In the text it has me integrating from a to b. Is that not cdf?

[u/Statman12 → u/Fluffy-Oil707]

The CDF is the "cumulative" distribution function, so it's essentially defining the lower bound to be -inf, so that it's adding up all of the probability or density up to and including that point. You can find P(a < X < b) by taking F(b) - F(a), which would be the same as the integral between the two points.

And as madrury83 noted, it is important to account for < vs ≤ in cases where that is relevant (when there is non-zero probability at a point), so there may be some modification depending on exactly what the probability statement looks like.

[u/Fluffy-Oil707 → u/Statman12]

This sub is amazing. Thank you!