Are the terms probability density function and probability distribution (or just "distribution") interchangeable?

Question

Like the title says, are the terms probability density function and probability distribution (or just "distribution") interchangeable? If not, what is the difference?

Answer 1

The phrase probability density function (pdf) means a specific thing: a function $f_X(\cdot)$ for a specific random variable $X$ (that's what that subscript there is for, to distinguish this function from the pdfs of other random variables) with the property that for all real numbers $a$ and $b$ such that $a < b$, $$P\{a < X \leq b\} = \int_a^b f_X(u)\,\mathrm du = \int_a^b f_X(v)\,\mathrm dv = \int_a^b f_X(t)\,\mathrm dt.$$ The different integrals are intended to serve as a reminder that it does not matter in the least what symbol we use as the argument of $f_X(\cdot)$ and that it is not the case (as is regrettably far too often believed by those starting on this subject) that the argument must be the lower-case letter corresponding to the upper-case letter that denotes the random variable. We also insist that $$\int_{-\infty}^\infty f_X(u)\,\mathrm du = 1.$$ If $P\{X = \alpha\} > 0$ for some real number $\alpha$, then $X$ does not have a pdf except for those who incorporate Dirac deltas into their probability calculus.

The cumulative probability distribution function (cdf or CDF) $F_X(\cdot)$ of $X$ is the function defined as $$F_X(\alpha) = P\{X \leq \alpha\}, -\infty < \alpha < \infty.$$ It is related to the pdf (for functions that do have a pdf) through $$F_X(\alpha) = \int_{-\infty}^\alpha f_X(u)\,\mathrm du.$$

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While there might be a very restrictive definition of the phrase probability distribution that some people insist on, the colloquial use of the term broadly encompasses the pdf and the CDF and the pmf (probability mass function which is also called the ddf or discrete density function) and whatever else we might want to include as descriptive of the probabilistic behavior of a random variable. For example, the phrase

the probability distribution of $X$ is uniform on $(a,b)$

will hardly ever be interpreted as meaning that the CDF of $X$ has constant value on $(a,b)~$!! Although it is the distribution which is alleged to be uniform, everyone in his/her right mind will take that as meaning that the density of $X$ has constant value $(b-a)^{-1}$ on the interval $(a,b)$ (and has value $0$ elsewhere). Similarly, for "$X$ is uniformly distributed on $(a,b)$" when what is meant is that the pdf of $X$ has constant value on $(a,b)$.

As another instance of colloquial usage of distribution to mean density, consider this quote from a recently posted answer by Moderator Glen_b.

"Saying the mode implies that the distribution has one and only one."

A density might possess a unique mode but a CDF cannot have a unique mode (in the unextended reals). However, no one reading that quote is likely to think that Glen_b meant the CDF when he wrote "distribution".

Answer 2

Distribution function is sometimes used to refer to the cumulative distribution function (CDF) $F_X$. $F_X(x)$ is a monotonically increasing function with $\lim_{x\to-\infty}F_X(x)=0$ and $\lim_{x\to+\infty}F_X(x)=1$.

Density function, in isolation, can mean a lot of things, but in the context of probability distributions, it often refers to the probability density function (PDF) $f_X$.

The difference between them is that, if $f_X$ is Lebesgue-integrable and non-negative everywhere:

$$F_X(x)=\int_{-\infty}^{x}f_X(u)du$$

If it's not, the probability distribution does not admit a density function, although every probability distribution admits a distribution function.

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See this for a more throughout discussion than I could offer: Are CDFs more fundamental than PDFs?

Answer 3

In terms of common usage, consider parsing the terminology used in R. The Description on the Distributions {stats} help page says:

Density, cumulative distribution function, quantile function and random variate generation for many standard probability distributions are available in the stats package.

For each of the built-in Distributions, it provides (according to the individual help pages) the "density" (e.g. dnorm for Normal, dbinom for Binomial) and the "distribution function" (e.g., pnorm, pbinom; called the "cumulative distribution function" on the main Distributions page, as quoted above).

So one might interpret that "probability distribution" describes (perhaps a member of) a family of distributions, "density" can be used for discrete distributions like the binomial, and the phrase "distribution function" might be preferred over "distribution" when the cumulative distribution function is what is intended.

Alternatively, one might argue that common usage even among the experienced often depends on context for clarity.

Answer 4

No.

1. "probability density function" is used only for continuous distributions. A discrete distribution can't have a pdf (though it can be approximated with a pdf). "probability distribution" is often used for discrete distributions, e.g., the binomial distribution.

2. "probability distribution" has a meaning for both discrete and continuous distributions, but a probability distribution is directly applicable only for discrete distributions. When the word is used with continuous distributions, it refers to an underlying mathematical construct such as the normal distribution, which must for most purposes be instantiated in a function, typically a probability density function or a cumulative density function, before it can be applied.