Probability distribution vs. probability mass function (PMF): what is the difference between the terms?

Consider a discrete case. PMF is the probability each value of random variable gets. So, for example, X ~ Poisson(2). I plot these probabilities (below), so I can say that I show the PMF of X. But on the other hand I show the distribution of X. For example, I can say whether the distribution I have is symmetrical or not. So, what is the difference between probability distribution and PMF terms (in discrete case)? Below I also bring the definitions from Wikipedia, but it is not helpful either.

Many thanks!

A probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value.

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

I'm not aware of an agreed upon definition/meaning for probability distribution.

On the other hand, probability mass functions and probability density functions have agreed upon definitions and are used to describe probability distributions.

A probability density function is the generalization of probability mass functions to random variables which are not strictly discrete. In the case of a discrete random variable, the main difference is that the probability density function should integrate to one, while the probability mass function should add to one.

Suppose $$X$$ is a discrete random variable taking values $$S=\{x_1,x_2,\ldots\} \subset \mathbb{R}$$.

The probability mass function is a function $$p : S\to [0,1]$$ where $$p(x) = \mathbb{P}(X=x)$$

On the other hand, the density function (of any RV) can be thought of as, $$f(x)dx = \mathbb{P}(X\in[x+dx])$$ In integral form you could write this as, $$\int_{x}^{x+dx} f(z)dz = \mathbb{P}(X\in [x,x+dx])$$

That is, the density times the width of a small interval gives the probability that $$X$$ is in that small interval $$X\in[x,x+dx]$$.

If the random variable is discrete, then the probability that $$X$$ is in this interval is the same as the probability $$X=x$$ for small enough $$dx$$. So you have $$f(x)dx = \mathbb{P}(X=x)$$ (or in integral form, $$\lim_{dx\to 0}\int_{x}^{x+dx} f(z)dz = \mathbb{P}(X=x)$$).

In particular, if $$p(x)$$ is the pmf for a discrete random variable $$X$$, then we can write the density function as: $$f(x) = \sum_{i:p(x_i)\neq 0} p(x_i) \delta(x-x_i)$$ where $$\delta(x)$$ is the delta distribution; i.e. $$\int_a^b f(x)\delta(c)d x = f(c)$$ whenever $$c\in[a,b]$$

• Only a continuous random variable has a density function. Note that $\lim_{dx \to 0} \int_x^{x+dx} f(z) dz = 0$, so your final equation states that $P(X = x) = 0$. This is true for a continuous random variable but not for a discrete random variable. (Discrete random variables don't have density functions.) – littleO Dec 31 '18 at 15:23
• You can write the density of discrete random variables using delta distributions. – tch Dec 31 '18 at 15:26

The word "distribution" gets thrown around loosely sometimes, which can cause confusion.

The distribution of a random variable $$X$$ is the function that takes a set $$S \subset \mathbb R$$ as input and returns the number $$P(X \in S)$$ as output. (Technically I should assume that $$S$$ is a "nice" subset of $$\mathbb R$$ in some sense, but let's not worry about that.) I think the Wikipedia article would be more clear if it just gave us this definition up front.

The PMF of a random variable $$X$$ is the function that takes a number $$x \in \mathbb R$$ as input and returns the number $$P(X=x)$$ as output. If $$X$$ is a discrete random variable, then the PMF of $$X$$ is a convenient way to specify the distribution of $$X$$.

Here is one way to describe the relationship between the distribution of $$X$$ and the PMF of $$X$$, in the case where $$X$$ is a discrete random variable. Suppose that the possible values of $$X$$ are $$x_1,x_2,\ldots$$ If $$f$$ is the distribution of $$X$$, then $$f(S) = \sum_{i : x_i \in S} P(X = x_i)$$ for any set $$S \subset \mathbb R$$.

• Thanks. But is not the definition of PMF by itself exactly as you described for "distribution of a random variable X"? If not, how they are different? – John Dec 31 '18 at 14:34
• No, those definitions are not the same. For example, the PMF of $X$ takes a number as input, but the distribution of $X$ takes a set of numbers as input. – littleO Dec 31 '18 at 14:38
• Thank you a lot, for clarification. But I am still a bit confused regarding your statement "the distribution of X takes a set of numbers as input". Suppose I flip a coin twice, X is number of tails. My PMF is: 0: 1/4, 1: 1/2, 2, 1/4. So, for each value X (0,1,2,) the PMF returns the P(X=x). How will look the distribution function? It would help you can update your answer. Thanks! – John Dec 31 '18 at 14:51
• Let $f$ be the distribution of the random variable $X$ that you mentioned. Then for example, if $S = (-7,1.3)$ (that is, $S$ is the open interval from $-7$ to $1.3$), then $f(S) = 3/4$. We could attempt to describe $f$ more explicitly but I'm not sure that it would be useful. The definition of $f$ is just that $f(S) = P(X \in S)$ for any set $S \subset \mathbb R$. – littleO Dec 31 '18 at 14:58
• OK. I see. Thanks a lot. – John Dec 31 '18 at 15:03

I provide a simple explanation of this here: Difference between "probability density function" and "probability distribution function"?. In short, a probability mass function is a discrete probability distribution function, where discrete is often implied.