A random variable is a function

that assigns a real number $X(u)$ to each outcome $u$ in the sample space $\mathrm{\Omega }$.

For every $t\in ℝ$, the set

is an event to which a probability can be assigned.

Let the sample space

A random variable on this sample space is

A random variable $X$ is called discrete if it takes values in a finite or countably infinite set $S$ with probability one:

The probability mass function of a discrete random variable $X$ is

Note: $p_X(k)=0$ for values outside the set of possible values of $X$.

Suppose that three students each have probability $1/5$ of arriving late, independently of one another. Let $X$ be the number of students who arrive late.

Then $X\in \{\mathrm{0,1,2,3}\}$, and

Unequivocally,

the probability that at least two students arrive late is

Let $X$ be a discrete random variable. Then:

1. For every $k\in S$,

   $$0\le p_X(k)\le 1.$$

2. The probabilities sum to one:

   $$\sum _k{p}_X(k)=1.$$

3. For $a,b\in ℝ$ with $a\le b$,

   $$\mathbb{P}(a\le X\le b)=\sum _{a\le k\le b}{p}_X(k).$$

   If $X$ is integer-valued and $a,b$ are integers, this becomes

   $$\mathbb{P}(a\le X\le b)=\sum _{k=a}^b{p}_X(k).$$

4. For every $a\in ℝ$,

   $$\mathbb{P}(X\le a)=\sum _{\begin{array}{c}k\le a\end{array}}{p}_X(k).$$

5. For every $a\in ℝ$,

   $$\mathbb{P}(X\ge a)=\sum _{\text{k}\ge a}{p}_X(k)=1-\mathbb{P}(X<a).$$

For any random variable $X$, the cumulative distribution function is defined by

This definition applies to both discrete and continuous random variables.

Let $F_X$ be the cumulative distribution function of a random variable $X$. Then $F_X$ is non-decreasing and right continuous. That is,

and

1. $$\underset{t\to \infty }{\lim }{F}_X(t)=1.$$
2. $$\underset{t\to -\infty }{\lim }{F}_X(t)=0.$$

3. For $a<b$,

   $$\mathbb{P}(a<X\le b)=F_X(b)-F_X(a).$$

4. For every $a\in ℝ$,

   $$\mathbb{P}(X>a)=1-F_X(a).$$

5. For every $a\in ℝ$,

   $$\mathbb{P}(X<a)=\underset{h\downarrow 0}{\lim }{F}_X(a-h)=F_X(a^{-}),$$

   where $F_X(a^{-})$ denotes the left limit of $F_X$ at $a$.

Suppose that a bus arrives exactly every ten minutes and that a passenger arrives at a unifor random time between two buses.

Let $X$ be the passengers waiting time measured in minutes. Then $X$ is uniformly distributed on $[\mathrm{0,10}]$, with density

For example,

A random variable $X$ is called absolutely continuous if there exists a non-negative measurable function

such that, for every $A\subseteq ℝ$,

The function $f_X$ is called a probability density function of $X$.

In these notes, ``continuous random variable’’ refers to a random variable with such a density.

If $X$ has a density, then for every fixed $t\in ℝ$,

Consequently, including or excluding the endpoints of an interval does not change its probability.

This does not apply to a constant random variable: if $X=c$ with probability one, then

If $f_X$ is a probability density function, then

and

Let $X$ be a discrete random variable with probability mass function $p_X$. Its expected value is

provided that

If $X$ has density $f_X$, its expected value is

provided that

The expected value dont have to be the most likely outcome, and it doesn’t need to be a value that $X$ can take.

For example, a random variable taking the values $0$ and $1$, each with probability $1/2$, has

even though $X$ never equals $1/2$.

Let $X$ be a random variable and let $g:ℝ\to ℝ$ be a function such that

Then:

1. If $X$ is discrete with probability mass function $p_X$,

   $$𝔼[g(X)]=\sum _{k\in S}g(k)p_X(k).$$

2. If $X$ has density $f_X$,

   $$𝔼[g(X)]={\int }_{-\infty }^{\infty }g(x)f_X(x)dx.$$

The variance of $X$, denoted by $V(X)$ or $\mathrm{Var}(X)$, is

The standard deviation of $X$ is

If $𝔼[X^2]<\infty$, then