Krdytkyu

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.

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 $S subset mathbb R$.

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 $p : Sto [0,1]$ where
$$
p(x) = mathbb{P}(X=x)
$$

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

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

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, $lim_{dxto 0}int_{x}^{x+dx} f(z)dz = mathbb{P}(X=x)$).

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 $cin[a,b]$