diff --git a/src/ainotes.cls b/src/ainotes.cls index 5312088..4370c4d 100644 --- a/src/ainotes.cls +++ b/src/ainotes.cls @@ -65,6 +65,7 @@ \renewcommand{\vec}[1]{{\mathbf{#1}}} \newcommand{\nullvec}[0]{\bar{\vec{0}}} \newcommand{\matr}[1]{{\bm{#1}}} +\newcommand{\prob}[1]{{\mathcal{P}({#1})}} \renewcommand*{\maketitle}{% diff --git a/src/statistical-and-mathematical-methods-for-ai/img/normal_distribution.png b/src/statistical-and-mathematical-methods-for-ai/img/normal_distribution.png new file mode 100644 index 0000000..17d59b5 Binary files /dev/null and b/src/statistical-and-mathematical-methods-for-ai/img/normal_distribution.png differ diff --git a/src/statistical-and-mathematical-methods-for-ai/main.tex b/src/statistical-and-mathematical-methods-for-ai/main.tex index 2356a8b..607f7ab 100644 --- a/src/statistical-and-mathematical-methods-for-ai/main.tex +++ b/src/statistical-and-mathematical-methods-for-ai/main.tex @@ -13,5 +13,6 @@ \input{sections/_matrix_decomp.tex} \input{sections/_vector_calculus.tex} \input{sections/_gradient_methods.tex} + \input{sections/_probability.tex} \end{document} \ No newline at end of file diff --git a/src/statistical-and-mathematical-methods-for-ai/sections/_probability.tex b/src/statistical-and-mathematical-methods-for-ai/sections/_probability.tex new file mode 100644 index 0000000..5498d57 --- /dev/null +++ b/src/statistical-and-mathematical-methods-for-ai/sections/_probability.tex @@ -0,0 +1,210 @@ +\chapter{Probability} + + +\section{Probability} +\begin{description} + \item[State space] \marginnote{State space} + Set $\Omega$ of all the possible results of an experiment. + \begin{example} + A coin is tossed two times. + $\Omega = \{ (\text{T}, \text{T}), (\text{T}, \text{H}), (\text{H}, \text{T}), (\text{H}, \text{H}) \}$ + \end{example} + + \item[Event] \marginnote{Event} + Set of possible results (i.e. $A$ is an event if $A \subseteq \Omega$) + + \item[Probability] \marginnote{Probability} + Let $\mathbb{E}$ be the set of all the possible events (i.e. power set of $\Omega$). + The probability is a function: + \[ \prob{A}: \mathbb{E} \rightarrow [0, 1] \] + \begin{example} + Let $\Omega$ be as above. + Given an event $A = \{ (\text{T}, \text{H}), (\text{H}, \text{T}) \}$, + its probability is: $\prob{A} = \frac{2}{4} = \frac{1}{2}$ + \end{example} + + \item[Conditional probability] \marginnote{Conditional probability} + Probability of an event $B$, knowing that another event $A$ happened: + \[ \prob{B \vert A} = \frac{\prob{A \cap B}}{\prob{A}} \text{, with } \prob{A} \neq 0 \] + + \begin{example} + A coin is tossed three times. + Given the events $A = \{ \text{tails two times} \}$ and $B = \{ \text{one heads and one tails} \}$ + We have that: + + \begin{minipage}{\linewidth} + \centering + \small + $\Omega = \{ + (\text{T}, \text{T}, \text{T}), (\text{T}, \text{T}, \text{H}), (\text{T}, \text{H}, \text{T}) + (\text{T}, \text{H}, \text{H}), (\text{H}, \text{T}, \text{T}), (\text{H}, \text{T}, \text{H}) + (\text{H}, \text{H}, \text{T}), (\text{H}, \text{H}, \text{H}) + \}$ + \end{minipage} + + \begin{minipage}{.325\linewidth} + \centering + $\prob{A} = \frac{4}{8} = \frac{1}{2}$ + \end{minipage} + \begin{minipage}{.325\linewidth} + \centering + $\prob{B} = \frac{6}{8} = \frac{3}{4}$ + \end{minipage} + \begin{minipage}{.325\linewidth} + \centering + $\prob{A \cap B} = \frac{3}{8}$ + \end{minipage} + + \begin{minipage}{.48\linewidth} + \centering + $\prob{A \vert B} = \frac{3/8}{3/4} = \frac{1}{2}$ + \end{minipage} + \begin{minipage}{.48\linewidth} + \centering + $\prob{B \vert A} = \frac{3/8}{1/2} = \frac{3}{4}$ + \end{minipage} + \end{example} + + \item[Independent events] \marginnote{Independent events} + Two events $A$ and $B$ are independent if: + \[ \prob{A \cap B} = \prob{A}\prob{B} \] + It follows that: + + \begin{minipage}{.48\linewidth} + \centering + $\prob{A \vert B} = \prob{A}$ + \end{minipage} + \begin{minipage}{.48\linewidth} + \centering + $\prob{B \vert A} = \prob{B}$ + \end{minipage} + + In general, given $n$ events $A_1, \dots, A_n$, they are independent if: + \[ \prob{A_1 \cap \dots \cap A_n} = \prod_{i=1}^{n} \prob{A_i} \] +\end{description} + + + +\section{Random variables} +\begin{description} + \item[Random variable (RV)] \marginnote{Random variable} + A random variable $X$ is a function: + \[ X: \Omega \rightarrow \mathbb{R} \] + + \item[Target space/Support] \marginnote{Target space} + Given a random variable $X$, + the target space (or support) $\mathcal{T}_X$ of $X$ is the set of all its possible values: + \[ \mathcal{T}_X = \{ x \mid x = X(\omega), \forall \omega \in \Omega \} \] +\end{description} + + +\subsection{Discrete random variables} + +\begin{description} + \item[Discrete random variable] \marginnote{Discrete random variable} + A random variable $X$ is discrete if its target space $\mathcal{T}_X$ is finite or countably infinite. + + \begin{example} + A coin is tossed twice. + + The random variable is $X(\omega) = \{ \text{number of heads} \}$. + We have that $\mathcal{T}_X = \{ 0, 1, 2 \}$, therefore $X$ is discrete. + \end{example} + + \begin{example} + Roll a die until 6 comes out. + + The random variable is $Y(\omega) = \{ \text{number of rolls before 6} \}$. + We have that $\mathcal{T}_Y = \{ 1, 2, \dots \} = \mathbb{N} \smallsetminus \{0\}$, + therefore $Y$ is discrete as $\mathcal{T}_Y$ is a countable set. + \end{example} + + \item[Probability mass function (PMF)] \marginnote{Probability mass function (PMF)} + Given a discrete random variable $X$, its probability mass function is a function $f_X: \mathcal{T}_X \rightarrow [0, 1]$ such that: + \[ f_X(x) = \prob{X = x}, \forall x \in \mathcal{T}_X \] + + A PMF has the following properties: + \begin{enumerate} + \item $f_X(x) \geq 0, \forall x \in \mathcal{T}_X$ + \item $\sum_{x \in \mathcal{T}_X} f_X(x) = 1$ + \item Let $A \subseteq \Omega$, $\prob{X = x \in A} = \sum_{x \in A} f_X(x)$ + \end{enumerate} + + \begin{example} + Let $\Omega = \{ (\text{T}, \text{T}), (\text{T}, \text{H}), (\text{H}, \text{T}), (\text{H}, \text{H}) \}$. + Given a random variable $X = \{ \text{number of heads} \}$ with $\mathcal{T}_X = \{ 0, 1, 2 \}$. + The PMF is: + \[ + \begin{split} + f_X &= \prob{X = 0} = \frac{1}{4} \\ + f_X &= \prob{X = 1} = \frac{2}{4} \\ + f_X &= \prob{X = 2} = \frac{1}{4} + \end{split} + \] + \end{example} +\end{description} + +\subsubsection{Common distributions} +\begin{descriptionlist} + \item[Uniform distribution] \marginnote{Uniform distribution} + Given a discrete random variable $X$ with $\#(\mathcal{T}_X) = N$, + $X$ has an uniform distribution if: + \[ f_X(x) = \frac{1}{N}, \forall x \in \mathcal{T}_X \] + + \item[Poisson distribution] \marginnote{Poisson distribution} + Given a discrete random variable $X$ with mean $\lambda$, + $X$ has a poisson distribution if: + \[ f_X(x) = e^{-\lambda} \frac{\lambda^x}{x!}, \forall x \in \mathcal{T}_X \] +\end{descriptionlist} + + +\subsection{Continuous random variables} + +\begin{description} + \item[Continuous random variable] \marginnote{Continuous random variable} + A random variable $X$ is continuous if its target space $\mathcal{T}_X$ is uncountably infinite (i.e. a subset of $\mathbb{R}$). + Usually, $\mathcal{T}_X$ is an interval or union of intervals. + + \begin{example} + Given a random variable $Z = \{ \text{Time before the arrival of a client} \}$. + $Z$ is continuous as $\mathcal{T}_Z = [a, b] \subseteq [0, +\infty[$ is an uncountable set. + \end{example} + + \item[Probability density function (PDF)] \marginnote{Probability density function (PDF)} + Given a continuous random variable $X$, + its probability density function is a function $f_X: \mathcal{T}_X \rightarrow \mathbb{R}$ such that: + \[ \prob{X \in A} = \int_{A} f_X(x) \,dx \] + \[ \prob{a \leq X \leq b} = \int_{a}^{b} f_X(x) \,dx \] + Note that $\prob{X = a} = \prob{a \leq X \leq a} = \int_{a}^{a} f_X(x) \,dx = 0$ + + A PDF has the following properties: + \begin{enumerate} + \item $f_X(x) \geq 0, \forall x \in \mathcal{T}_X$ + \item $\int_{x \in \mathcal{T}_X} f_X(x) \,dx = 1$ + \item $\prob{X \in A} = \int_{A} f_X(x) \,dx$ + \end{enumerate} +\end{description} + +\subsubsection{Common distributions} +\begin{descriptionlist} + \item[Continuous uniform distribution] \marginnote{Continuous uniform distribution} + Given a continuous random variable $X$ with $\mathcal{T}_X = [a, b]$, + $X$ has a continuous uniform distribution if: + \[ f_X(x) = \frac{1}{b-a}, \forall x \in \mathcal{T}_X \] + + \item[Normal distribution] \marginnote{Normal distribution} + Given a continuous random variable $X$ and the parameters $\mu$ (mean) and $\sigma$ (variance). + $X$ has a normal distribution if: + \[ f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{\frac{-(x-\mu)^2}{2\sigma^2}} , \forall x \in \mathcal{T}_X\] + + \begin{description} + \item[Standard normal distribution] \marginnote{Standard normal distribution} + Normal distribution with $\mu = 0$ and $\sigma = 1$. + \end{description} + + \begin{figure}[ht] + \centering + \includegraphics[width=0.5\textwidth]{img/normal_distribution.png} + \caption{Normal distributions and standard normal distribution} + \end{figure} +\end{descriptionlist} \ No newline at end of file