\chapter{Random variables}


In the advanced part of the book, we will be able to always develop methods that work in both the sense of discrete and continuous random variables, and the distinction of the theory of discrete and continuous random variables will cease to exist.


Memoryless RV
Symmetric RV


\section{Expectation}
- Expectation
- Markov's inequality
\section{Variance}
- Chebyshev's inequality
- Cantelli's inequality

\section{Discrete probability distributions}


There are many important probability distributions, so much so that it warrants a whole body of literature in itself. Consequently, in this book we will discuss the most fundamental and frequently encountered probability distributions.

Bernoulli
\[X : \mathcal{F} \to \{0,1\} \]
\[f_X (n) = \begin{cases} p & n=1 // 1-p & n=0 \end{cases} \]

Binomial
\[X : \mathcal{F} \to \mathbb{N} \cap [0,n] \]
\[f_X (k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Geometric
\[X : \mathcal{F} \to \mathbb{N} \setminus \{0\} \]
\[f_X (n) = (1-p)^{n-1}p \]

Poisson
\[X : \mathcal{F} \to \mathbb{N}\]
\[f_X (n) = \frac{e^{-\lambda} \lambda^n }{n!} \]


\section{Continuous probability distributions}

Uniform
Exponential
Normal


\section{Theorems on RVs}
\[\mathrm{Pr}(X=n) = \sum_{m \in X(\mathcal{F})} \mathrm{Pr}(X=n|)\mathrm{Pr}(F) \]
- Law of total probability
- Law of total expectation
- Law of total variance

There are 2 more noteworthy laws;  the \emph{law of large numbers} and the \emph{central limit theorem}, which have major applications in statistics. We mention them so that they can be used in applications, however their proofs are defered for the advanced part of this book.


- Law of large numbers (LLN)
- Central limit theorem (CLT)


\section{Probability generating functions}






\section{Kurtosis}
\section{Skewness}
\section{Moments}


- Random variable (RV)
- PMF/PDF
- CDF
- MGF/CF
- Expectation
- Variance
- Kurtosis
- Skewness
- Moments
- Probability distribution
- Independence
- Correlation
- Covariance