Random Variable

def. Random Variable A Random Variable $X$ from probability space $(\Omega ,\mathcal{F},\mathbb{P})$ to measurable space $(\mathbb{R},\mathcal{B})$ is a function:

$$X:\Omega \to \mathbb{R}$$

• $\mathcal{B}$ is the Borel sigma-algebra on real numbers (=Borel set)
  • This is a set of open intervals on $\mathbb{R}$ that satisfies $\sigma$-algebra properties
• for all $B\in \mathcal{B}$, it is true that $X^{-1}(B)=\{\omega \in \Omega \mid X(w)\in B\}\in \mathcal{F}$
  • You’re given an open interval (=set) $B$ on real numbers
  • Get all $\omega$ such that $X(\omega )$ is within this range (=pre-image)
  • This set (a subset of $\Omega$) must be in the sigma-algebra of $\Omega$
  • This applies to any range $B=(a,b)$
  • & We can then say ”$X$ is $\mathcal{F}$-measurable” or ”$\mathcal{F}$ has enough information to measure $X$“.
  • Intuition. Let there be other sigma-algebras $\mathcal{G},\mathcal{H}$ such that $\mathcal{G}\subset \mathcal{F}\subset \mathcal{H}$, in addition to $X$ being $\mathcal{F}$-measurable. In this case…
    • $X$ is $\mathcal{H}$-measurable because $\mathcal{H}$ has more information than $\mathcal{F}$.
    • However, $X$ is not $\mathcal{G}$-measurable because $\mathcal{G}$ has less information than $\mathcal{F}$.

def. Probability on a Random Variable. Probability function on random variable $X$ is a function ${\mathbb{P}}_X:\mathcal{R}\to [0,1]$ such that:

$${\mathbb{P}}_X(B):=\mathbb{P}(X^{-1}(B))=\mathbb{P}(\{\omega \in \Omega \mid X(\omega )\in A\})$$

• & i.e., the probability we encounter daily $\mathbb{P}(X\dots )$ is simply a shorthand notation ${\mathbb{P}}_X=\mathbb{P}\circ X^{-1}$
• You put the interval $B$, and get the probability of all $\omega$ in that interval’s pre-image
• thus by abuse of notation we write: ${\mathbb{P}}_X(B)\equiv \mathbb{P}(X\in B)\equiv \mathbb{P}(X^{-1}(B))$

Example. Let $\Omega$ is the trajectory of a coin toss. This is a very big set, and probably also infinite. On the other hand, let $X$ a random variable for

$$X=\{\begin{array}{ll}1& \text{if heads}\\ 0& \text{if tails}\end{array}$$

• • This is much simpler and more useful.
• This means $X^{-1}(1)=\{\text{All trajectories that land in heads}\}$.
• If $X^{-1}\in \mathcal{F}$, then we can try to measure the probability of $\mathbb{P}(X^{-1}(1))=\mathbb{P}(\{\text{All trajectories that land in heads}\})$
• Because we know the pre-image of $1$ is in $\mathcal{F}$, we know that $\mathbb{P}(X^{-1}(1))$ is defined.

thm. Addition Rule for Random Variables. For a discrete random variable $X$:

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

Functions of Random Variables §

Motivation. Functions can be made of random variables; for example let $Y=|X-1|$. In order to investigate $Y$, need a way to derive the probability distribution of $Y$ from $X$.

Example. Let $Y$ be a random variable defined by a function of another random variable $X$; $Y=f(X)$. Then:

$$\mathbb{P}(Y=y)=\mathbb{P}(f(X)=y)=\sum _{\text{all x s.t. f(x)=y}}\mathbb{P}(X=x)$$

thm. Random variables $X,Y$ are equal when:

1. $\text{Range}(X)=\text{Range}(Y)$
2. ${\forall }_{k\in \text{Range}}\mathbb{P}(X=k)=\mathbb{P}(Y=k)$

Indicator Functions §

def. Indicator Functions. For event $A\subset \Omega$, the indicator function ${\mathbb{I}}_A$ is a random variable (i.e. function) such that:

$$\begin{gathered} {\mathbb{I}}_A:\Omega \to \{0,1\} \\ \text{s.t.} \\ {\mathbb{I}}_A(\omega )=\{\begin{array}{ll}0& \omega \notin A\\ 1& \omega \in A\end{array} \end{gathered}$$

Remark.

• Indicator functions are useful in probability for solving problems, not for being a fundamental mathematical object.
• Remember that indicator functions are also random variables. All the rules for random variables apply, including the identities for expected values.

Properties. let $\mathbb{I}$ an indicator function describing an event with probability $p$, then:

• $\mathbb{E}(\mathbb{I})=p$
• $\text{Var}(\mathbb{I})=p(1-p)$