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\begin{document}

MATH 3033 based on

Dekking et al. A Modern Introduction to Probability and Statistics. 2007

Slides by Solomon Feitelson 

Instructor Longin Jan Latecki

{\bf \large C4: DISCRETE\_RANDOM\_VARIABLES}

\lyxline{\normalsize}

DEFINITION OF A DISCRETE RANDOM VARIABLE

Let $\Omega$ be an arbitrary sample space. We can have 2 types of
sample spaces:

0) FINITE: $\Omega=\left\{ a_{1},a_{2},\cdots,a_{n}\right\} $

1) INFINITE: $\Omega=a_{1},a_{2},\cdots$

As $\Omega$ is arbitrary, it can assume any arbitrary collection
of arbitrary values.

$\newline$A discrete random variable $X$ is a function $X:\Omega\rightarrow\mathbb{R}$.

$X$ receives an arbitrary sample space $\Omega$ as input, and maps
$\Omega$ to a set of Real Numbers $\mathbb{R}$.

\lyxline{\normalsize}

DEFINITION OF A PROBABILITY MASS FUNCTION

The Probability Mass Function $p$ of a discrete random variable X
is the function $p:\mathbb{R}\rightarrow\left[0,1\right]$.

$p$ receives a set of Real Numbers $\mathbb{R}$ as input, and maps
the set to the inclusive interval $\left[0,1\right]$.

We define $p$ as:

\[
p\left(a\right)=P\left(\left\{ X=a\right\} \right)\mid-\infty<a<\infty.\]


$\newline$Explanation of the formula:

0) Since $\Omega$ is arbitrary, value $a$ assumes an arbitrary range.

1) Set $\left\{ X=a\right\} \subseteq\Omega$ describes an event or
events that occur in $\Omega$, for a particular $a$.

2) $P\left(\left\{ X=a\right\} \right)$ is the probability that a
particular $a$ will occur in $\Omega$.

$\newline$If X assumes a finite number of values $\left\{ a_{1},a_{2},\cdots,a_{n}\right\} $,
then:

0) $p\left(a_{i}\right)>0$.

1) $p\left(a_{1}\right)+p\left(a_{2}\right)+\cdots+p\left(a_{n}\right)=1$.

2) $p\left(a\right)=0$ for all other values of $a$.

$\newline$If X assumes an infinite number of values $a_{1},a_{2},\cdots$,
then:

0) $p\left(a_{i}\right)>0$.

1) $p\left(a_{1}\right)+p\left(a_{2}\right)+\cdots=1$.

2) $p\left(a\right)=0$ for all other values of $a$.

\lyxline{\normalsize}

DEFINITION OF A CUMULATIVE/DISTRIBUTION FUNCTION

The Cumulative/Distribution Function F of a random variable X is the
function $F:\mathbb{R}\rightarrow\left[0,1\right]$.

$F$ receives a set of Real Numbers $\mathbb{R}$ as input, and maps
the set to the inclusive interval $\left[0,1\right]$.

We define $F$ as:

\[
F\left(a_{w}\right)=P\left(\left\{ X\leq a_{w}\right\} \right)\mid-\infty<a_{w}<\infty.\]


$\newline$Another valid definition of $F$ is:

\[
F\left(a_{w}\right)=\sum{a_{w}\atop k=a_{1}}\left[p\left(k\right)\right].\]


$\newline$Explanation of the formulas:

0) Since $\Omega$ is arbitrary, $a_{w}$ assumes an arbitrary range.

1) Set $\left[\left\{ X\leq a_{w}\right\} \subseteq\Omega\right]\mid\left[\forall k\in X,\left[k\leq a_{w}\right]\right]$.

2) $P\left(\left\{ X\leq a_{w}\right\} \right)$ is the cumulative
probability of all such events $k\in X$.

\lyxline{\normalsize}

DEFINITION OF A BERNOULLI DISTRIBUTION

A discrete random variable X has a Bernoulli Distribution with these
parameters:

Success Probability $p\mid0\leq p\leq1$.

Failure Probability $1-p\mid0\leq1-p\leq1$.

$\newline$We define the Probability Mass Function $p_{x}$ of a Bernoulli
Distribution as:

\[
p_{x}\left(1\right)=P\left(\left\{ X=1\right\} \right)=p.\]
\[
p_{x}\left(0\right)=P\left(\left\{ X=0\right\} \right)=1-p.\]


$\newline$Explanation of the formulas:

0) A Bernoulli Distribution is suitable to model experiments with
only 2 possible outcomes:$\newline$

\[
1=\text{\textquotedblright}success\text{\textquotedblright};0=\lyxmathsym{\textquotedblright}failure\lyxmathsym{\textquotedblright}.\]


1) $p_{x}\left(1\right)=P\left(\left\{ X=1\right\} \right)$ is the
probability that the event $1="success"$ shall occur.

2) $p_{x}\left(0\right)=P\left(\left\{ X=0\right\} \right)$ is the
probability that the event $0="failure"$ shall occur.

$\newline$We denote this Bernoulli Distribution as $Ber\left(p\right)$.

\lyxline{\normalsize}

DEFINITION OF A BINOMIAL DISTRIBUTION

A discrete random variable X has a Binomial Distribution with these
parameters:

$n\mid n=1,2,\cdots$

1x Success Probability $p\mid0\leq p\leq n.$

1x Failure Probability $1-p\mid0\leq1-p\leq n.$

$\newline$We define the Probability Mass Function $p_{x}$ of a Binomial
Distribution as:

\[
p_{x}\left(k\right)=P\left(\left\{ X=k\right\} \right)=\left[{n\atop k}\right]\left[p^{k}\right]\left[\left[1-p\right]^{\left[n-k\right]}\right]\mid k=0,1,\cdots,n.\]


$\newline$Explanation of the formula:

0) $\left[{n\atop k}\right]=\frac{n!}{k!\left[n-k\right]!}$: Combination: from
n elements, the number of$\newline$order-insensitive ways to choose
k elements.

1) $\left[p^{k}\right]$: The probability of a {}``success,'' repeated
$k$ times.

2) $\left[\left[1-p\right]^{\left[n-k\right]}\right]$: The probability
of a {}``failure,'' repeated $\left[n-k\right]$ times.

$\newline$We denote this Binomial Distribution by $Bin\left(n,p\right)$.

\lyxline{\normalsize}

DEFINITION OF A GEOMETRIC DISTRIBUTION

A discrete random variable X has a Geometric Distribution with these
parameters:

1x Success Probability $p\mid0\leq p\leq1.$

1x Failure Probability $1-p\mid0\leq1-p\leq1.$

$\newline$We define the Probabillity Mass Function $p_{x}$ of a
Geometric Distribution as:

\[
p_{x}\left(k\right)=P\left(\left\{ X=k\right\} \right)=\left[\left[1-p\right]^{\left[k-1\right]}\right]p\mid\left[k=1,2,\cdots\right].\]


$\newline$Explanation of the formula:

0) $\left[k\right]$: The number of {}``repetitions'' of the experiment,
until a {}``success'' occurs.

1) $\left[\left[1-p\right]^{\left[k-1\right]}\right]$: The probability
of a {}``failure,'' repeated $\left[k-1\right]$ times.

2) $\left[p\right]$: The probability of a {}``succcess,'' repeated
1 time, after the execution of $\left[k-1\right]$ {}``failures.''
\end{document}
