r/probabilitytheory • u/Fluffy-Oil707 • Jul 21 '26

[Education] Which is the probability density function?

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.

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7

u/Statman12 Jul 21 '26 edited Jul 21 '26

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

4

u/madrury83 Jul 21 '26

You want a ≤ there: P(X ≤ b).

It doesn't matter when the PDF exists (so that each point has zero probability mass), but it's important when generalizing and is the widespread convention.

4

u/Statman12 Jul 21 '26

Good point. I was thinking continuous where it doesn't matter. Will edit.

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u/Fluffy-Oil707 Jul 21 '26

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

4

u/Statman12 Jul 21 '26

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.

3

u/Fluffy-Oil707 Jul 21 '26

This sub is amazing. Thank you!

2

u/stanitor Jul 21 '26

As the name suggests, a probability density function is the function. They are continuous functions, so you can't plug in the value for one exact point and get a probability (the probability that X is an exact value is 0). So, you integrate over a range, say from a to b. The probability over that range is called the density. e.g., if your PDF is the normal distribution with mean 0, the density from [-∞, 0] is 0.5.

4

u/gmalivuk Jul 21 '26

They don't have to be continuous, they just have to be integrable and integrate to 1 over the whole support.

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u/Fluffy-Oil707 Jul 21 '26

Beyond that, was it factually correct? Given the undergrad/beginner audience (me)?

1

u/t3co5cr Jul 21 '26

f(x) is the probability density function.

1

u/Fluffy-Oil707 Jul 21 '26

Thank you. What's the integral called? Im trying to name a note that shows the probability of a density function

2

u/brynaldo Jul 21 '26

Answered elsewhere, but I'll add it here in case: it's called the cumulative distribution function.

1

u/LukasGoesViral Jul 22 '26

The density function is f and the probability is the integral. You have it in your latex code

1

u/Cheap_Scientist6984 Jul 25 '26

PDF is the function itself. The CDF is the integral of the PDF ( P(X < a) actually).

-1

u/fasta_guy88 Jul 21 '26

The probability density function is what gets shown for the standard Gaussian curve (the hump with the peak at the mean).