3  Random variables

Definition 3.1 ($\mathbf{\text{Random variable}}$)
Let’s consider a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, then a random variable is a map where $({\mathrm{\Omega }}^{\prime },{\mathcal{B}}^{\prime })=(\mathbb{R},\mathcal{B}(\mathbb{R}))$ and therefore the map takes values on the real line, i.e. $$X:(\mathrm{\Omega },\mathcal{B})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$$ and such that: $$\mathrm{\forall }B\in \mathcal{B}(\mathbb{R})X^{-1}(B)=\{\omega \in \mathrm{\Omega }:X(\omega )\in B\}\subset \mathcal{B}$$

Note that when $X$ is a random variable the test for measurability (Definition 2.3), became: $$X^{-1}((-\mathrm{\infty },y])=[X(\omega )\le y]\subset \mathcal{B}\mathrm{\forall }y\in \mathbb{R}$$

Discrete vs continuous random variables

Definition 3.2 Let $X$ be a random variable with set of possible outcomes ${\mathrm{\Omega }}^{\prime }$. Then, $X$ is called discrete random variable if ${\mathrm{\Omega }}^{\prime }$ is either a finite set or a countably infinite set. $X$ is called continuous random variable if ${\mathrm{\Omega }}^{\prime }$ is either a an uncountable infinite set.

3.1 Induced distribution function

Consider a probability space $(\mathrm{\Omega },\mathcal{B},\mathbb{P})$ and a measurable map $X:(\mathrm{\Omega },\mathcal{B})\to ({\mathrm{\Omega }}^{\prime },{\mathcal{B}}^{\prime })$, then the composition $\mathbb{P}\circ X^{-1}$ is again a map. In this way at each element $\omega \in \mathrm{\Omega }$ is attached a probability measure. In fact, the composition is a map such that $$\mathbb{P}\circ X^{-1}:({\mathrm{\Omega }}^{\prime },{\mathcal{B}}^{\prime })\to [0,1]⟺({\mathrm{\Omega }}^{\prime },{\mathcal{B}}^{\prime })\stackrel{X^{-1}}{⟶}(\mathrm{\Omega },\mathcal{B})\stackrel{\mathbb{P}}{⟶}[0,1]$$ In general, the probability of a subset $A^{\prime }\in {\mathcal{B}}^{\prime }$ is denoted equivalently as: $$\mathbb{P}\circ X^{-1}(A^{\prime })=\mathbb{P}(X^{-1}(A^{\prime }))=\mathbb{P}(X(\omega )\in A^{\prime })$$

Example: Map and inverse map (continued)

Example 3.1 Let’s continue from the Example 2.1 and compute the probability of $\mathbb{P}\circ X^{-1}(\{+1\})$. Let’s consider one random extraction from the 52 cards, then for each distinct number we have 4 copies. Therefore the probability is computed as: $$\begin{array}{rl}\mathbb{P}(X^{-1}(\{+1\}))& =\mathbb{P}(\{\omega \in \mathrm{\Omega }:X(\omega )\in \{-1\}\})=\\ & =\mathbb{P}(\{2,3,4,5,6\})=\\ & =\frac{5\cdot 4}{52}=\frac{5}{13}\approx 38.46\mathrm{\%}\end{array}$$ Let’s now consider the probability of observing either $\{+1\}$ or $\{-1\}$, then $$\begin{array}{rl}\mathbb{P}(X^{-1}(\{-1,+1\}))& =\mathbb{P}(\{\omega \in \mathrm{\Omega }:X(\omega )\in \{-1,+1\}\})=\\ & =\mathbb{P}(\{2,3,4,5,6,10,11,12,13,14\})=\\ & =\frac{10\cdot 4}{52}=\frac{10}{13}\approx 76.92\mathrm{\%}\end{array}$$ Finally, by property of the probability measure $\mathbb{P}(X(\omega )\in \{0\})=1-\mathbb{P}(X(\omega )\in \{-1,+1\})\approx 23.08\mathrm{\%}$.

3.1.1 Distribution function on $\mathbb{R}$

When $X$ is a random variable the composition $\mathbb{P}(X^{-1}(A^{\prime }))$ is a probability measure induced on $\mathbb{R}$ by the distribution: $$\mathbb{P}(X^{-1}((-\mathrm{\infty },y])=\mathbb{P}(X\le y)\mathrm{\forall }y\in \mathbb{R}$$ Hence, it is possible to attach to a random variable a distribution function of $X$ that is a measure induced on the real line $\mathbb{R}$ and defined as: $$F_X(y)=\mathbb{P}(X(\omega )\in [-\mathrm{\infty },y))=\mathbb{P}(X\le y)$$ The distribution of $X$ is a function that goes from $F_X:(\mathbb{R},\mathcal{B}(\mathbb{R}))\to [0,1]$. If a random variable has a continuous and it’s distribution function is differentiable, then it is possible to define the density as: $$\begin{array}{}\text{(3.1)}& f_X(y)=\frac{d{F}_X(y)}{dy}⟺d{F}_X(y)=f_X(y)dy\end{array}$$