# Difference between density and distribution [in formal mathematical terms]

A similar question has been already asked but its not in mathematical framework and therefore seems to be different. According to definitions from the book that I am reading, a random variable and a distribution are defined as follows:

Definition. Let $(\Omega', \mathcal{A}')$ be a measurable space and let $X:\Omega\to\Omega'$ be measurable. Then $X$ is called a random variable.

Definition. Let $X$ be a random variable. The probability measure $P_X:=P\circ X^{-1}$ is called the distribution.

Now according to what I see in physical textbooks there is some other thing called density and that differs from distribution. How that one is formally defined?

• Short answer: a probability density, when it exists, is the derivative of its corresponding distribution. – Giuseppe Negro Apr 6 '14 at 16:24
• @GiuseppeNegro, Thank you. It would have been great if you have written this as an answer – Cupitor Apr 6 '14 at 17:57

The distribution is simply the assignment of probabilities to sets of possible values of the random variable. If I tell you how probable it is that a certain random variable is between $3$ and $5$, and also how probably it is that it's in every other possible set, then I've told you the distribution. Since I can't do this for every set individually, since there are infinitely many sets, perhaps a more down-to-earth way to say this is this: Suppose $X$ and $Y$ are random variables. If it is true of every set that the probability that $X$ is in that set is the same as the probability that $Y$ is in that same set, then $X$ and $Y$ have the same distribution.

A probability density function is a way of characterizing some distributions. For example, consider the function $$f(x) = \begin{cases} 0 & \text{if }x<0, \\ e^{-x} & \text{if }x\ge 0. \end{cases}$$ To say that this is the probability density function of a random variable $X$ is to say that for every measurable set $A$ of real numbers, $$\Pr(X\in A) = \int_A f(x)\,dx.$$ The probability assigned to each set $A$ is given by the integral above. A more concrete example: $$\Pr(3<X<5) = \int_3^5 e^{-x}\,dx\text{ and }\Pr(X\ge 2) = \int_2^\infty e^{-x}\,dx.$$

Not every probability distribution has a density. Say we let $X$ be the number of aces when a die is thrown four times. Then $X\in\{0,1,2,3,4\}$. The probability distribution assigns a positive number to every set that intersects that last set. For example the set $\{x : x\ge 3.2\}$ intersects $\{0,1,2,3,4\}$ and thus the probability distribution of $X$ assigns a positive number to that set. But there is no function $f$ such that for every set $A$ we have $\int_A f(x)\,dx$ equal to the probability that $X\in A$.

PS prompted by comments below: To put it in a different kind of language: Say $m$ is a measure (not necessarily assigning finite measure to the whole space) on the set of all measurable subsets of a space $S$. A probability density with respect to the measure $m$ is a measurable function $f:S\to[0,\infty)$ such that the function $$A\mapsto \int_A f\,dm$$ is a probability measure on the set of measurable subsets of $S$.

A probability distribution on $S$ is simply a probability measure on the set of all measurable subsets of $S$. But not quite "simply": The probability distribution of a random variable $X:\Omega\to S$ is the probability measure on measurable subsets of $S$ that assigns measure $P(\{\omega\in\Omega : X(\omega)\in A\})$ to each measurable subset $A$ of $S$.

PPS: When $f\ge0$ is a measurable function on Borel or Lebesgue-measurable subsets of $\mathbb R$, one sometimes refers to the "measure" $f(x)\,dx$, meaning the measure $$A\mapsto \int_A f(x)\,dx.$$ If in addition $\displaystyle\int_{\mathbb R} f(x)\,dx=1$, so that $f$ is a probability density, then one may similarly refer to the "probability distribution" $f(x)\,dx$.

(Of course, not all probability distributions on Borel subsets of the real line are of this form.)

• I appreciate the time you have spent on answering the question with such a detailed answer but I was looking for the formal definition. One more thing is what you have defined again(distributions) has been already defined as part of my question. – Cupitor Apr 6 '14 at 15:47
• @Cupitor : I've now added a "formal" definition. – Michael Hardy Apr 6 '14 at 16:17
• @O.B.D.A. : Typo fixed. – Michael Hardy Apr 6 '14 at 16:45