We discuss the properties of the probability distribution of a random variable. It is a measure of probability used in probability theory.

**What is a random variable?**

Consider a probability space $(\Omega,\mathscr{A},\mathbb{P})$. We also denote by $\mathscr{B}$ the Borel algebra defined by the open sets of $\mathbb{B}$. So a set $B\in\mathscr{B}$ is called a Borel set.

We say that $X\mapsto \mathbb{R}$ is a random variable if for any $B\in \mathscr{B},$ the event $$\{\omega\in\Omega:X(\omega)\in\mathscr{A}\}.$$ Formally, this definition means that the values of the random variable is corresponding to the outcomes of the random experiment.

Throughout this post, we use the following notation $$(X^{-1}(B))=(X\in B):=\{\omega\in\Omega:X(\omega)\in\mathscr{A}\}.$$ The sum, the product of radom variables are random variables.

**The probability distribution of a random variable **

According to the previous paragraph, if $B\in\mathscr{B}$ is a Borel set, then $X^{-1}(B)\in \mathscr{A},$ is an event. So that the probability $\mathbb{P}(X^{-1}(B))$ is well-defined. This allows us to set the following concept.

**Definition:** Consider a random variable $X:(\Omega,\mathscr{A})\to(\mathbb{R},\mathscr{B})$. We define a probability measure associated with $X$ by $(\mathbb{R},\mathscr{B})\to \mathbb{R}$ and \begin{align*}\mathbb{P}_X (B)=\mathbb{P}(X^{-1}(B)),\qquad \forall B\in \mathscr{B}.\end{align*} The probability $\mathbb{P}_X$ is called the probability distribution of the random variable $X$.

Notice that in several situations the probability distribution $\mathbb{P}_X$ replaces the initial probability $\mathbb{P}$ in the sense that the initial probability space $(\Omega,\mathscr{A},\mathbb{P})$ remains in the background, hidden; it is replaced with the more appropriate measure space $(\mathbb{R},\mathscr{R},\mathscr{P}_X)$.

We the random variable $X$ is discrete, that is, $X(\Omega)\subset \mathbb{N},$ then the probability distribution of the discrete random variable $X$ is $p_n=\mathbb{P}(X=n)$. Here $(X=n)=\{ \omega\in \Omega: X(\omega)=n\}$. This notion is used in elementary probability courses.

**The image of a random variable by a real measurable function**

Let $\psi:(\mathbb{R},\mathscr{B})\to (\mathbb{R},\mathscr{B})$ be a measurable function; in the sense that $\psi^{-}(B)\in\mathscr{B}$ for any Borel set $B\in \mathscr{B}$. Now if $X:(\Omega,\mathscr{A})\to(\mathbb{R},\mathscr{B})$ is a random variable, then the expectation of the new random variable $f(X)$ is given by \begin{align*}\mathbb{E}(\psi(X))=\int^{+\infty}_{-\infty} \psi(x)d\mathbb{P}_X(x).\end{align*} This result is known as the transfer theorem.

**Some known probability distributions **

In this section, we list some classical probability distributions:

**Bernoulli distribution** **of parameter** $p\in [0,1]$: it is associated with the discrete random variable $X$ such that $X(\Omega)=\{0,1\}$ such that $\mathbb{P}(X=0)=1-p$ and $\mathbb{P}(X=1)=p$. We only have two possibilities “success” when $X=1$ and “failure” when $X=0$. In this case we write $X\in \mathcal{B}(p)$.

**Binomial distribution of parameters $n\in\mathbb{N}$ and $p\in [0,1]$:** A random variable $X$ have this kind of distribution if it is of the form $X=X_1+\cdots+X_n$, where $X_i$ are independent random variables and $X_i\in \mathcal{B}(p)$ for any $i=1,\cdots,n$. In this case, we write $X\in \mathcal{B}(n,p)$ and we have \begin{align*} \mathbb{P}(X=k)=(\begin{smallmatrix}n\\ k\end{smallmatrix})p^k(1-p)^{1-k}.\end{align*} here the Binomial coefficients are defined by $$ (\begin{smallmatrix}n\\ k\end{smallmatrix})=\frac{n!}{k!(n-k)!}.$$