#import "./dvd.typ": *

#show: dvdtyp.with(
  title: "Probability and Statistics",
  author: "Youwen Wu",
)

#outline()

= Lecture 1

== Preliminaries

#definition("Statistics")[
  The science dealing with the collection, summarization, analysis, and
  interpretation of data.
]

== Set theory for dummies

A terse introduction to elementary set theory and the basic operations upon
them.

#definition[Set][
  A collection of elements.
]

#example[Examples of sets][
  + Trivial set: ${1}$
  + Empty set: $emptyset$
  + $A = {a,b,c}$
]

We can construct sets using set-builder notation (also sometimes called set comprehension).

$ {"expression with" x | "conditions on" x} $

#example("Set builder notation")[
  + The set of all even integers: ${2n | n in ZZ}$
  + The set of all perfect squares in $RR$: ${x^2 | x in NN}$
]

We also have notation for working with sets:

With arbitrary sets $A$, $B$, $Omega$:

+ $a in A$ ($a$ is a member of the set $A$)
+ $a in.not A$ ($a$ is not a member of the set $A$)
+ $A subset.eq Omega$ (Set theory: $A$ is a subset of $Omega$) (Stats: $A$ is a sample space in $Omega$)
+ $A subset Omega$ (Proper subset: $A != Omega$)
+ $A^c$ or $A'$ (read "complement of $A$")
+ $A union B$ (Union of $A$ and $B$. Gives a set with both the elements of $A$ and $B$)
+ $A sect B$ (Intersection of $A$ and $B$. Gives a set consisting of the elements in *both* $A$ and $B$)
+ $A \\ B$ (Set difference. The set of all elements of $A$ that are not also in $B$)
+ $A times B$ (Cartesian product. Ordered pairs of $(a,b)$ $forall a in A$, $forall b in B$)

We can also write a few of these operations precisely as set comprehensions.

+ $A subset Omega => A = {a | a in Omega, forall a in A}$
+ $A union B = {x | x in A or x in B}$ (here $or$ is the logical OR)
+ $A sect B = {x | x in A and x in B}$ (here $and$ is the logical AND)
+ $A \\ B = {a | a in A and a in.not B}$
+ $A times B = {(a,b) | forall a in A, forall b in B}$

Convince yourself that these definitions are equivalent to the previous ones.

#example[The real plane][
  The real plane $RR^2$ can be defined as a Cartesian product of $RR$ with itself.

  $ RR^2 = RR times RR $
]

Check your intuition that this makes sense. Why do you think $RR^n$ was chosen as the notation for $n$ dimensional spaces in $RR$?

#remark[Disjoint sets][
  If $A sect B$ = $emptyset$, then we say that $A$ and $B$ are *disjoint*.
]

#theorem[Properties of set operations][
  + DeMorgan's Laws:
    + $(A union B)' = A' sect B'$
    + $(A sect B)' = A' union B'$
]

#remark[Generalized DeMorgan's][
  + $(union_i A_i)' = sect_i A_i'$
  + $(sect_i A_i)' = union_i A_i'$
]

=== Sizes of infinity

#definition("Cardinality")[
  Let $N(A)$ be the number of elements in $A$. $N(A)$ is called the _cardinality_ of $A$.
]

Sets are either finite or infinite. Finite sets have a fixed finite cardinality.

Infinite sets can be either _countably infinite_ or _uncountably infinite_.

When a set is countably infinite, its cardinality is $aleph_0$ (here $aleph$ is
the Hebrew letter aleph and read "aleph null").

When a set is uncountably infinite, its cardinality is greater than $aleph_0$.

#example("Countable sets")[
  + The natural numbers $NN$.
  + The rationals $QQ$.
  + The natural numbers $ZZ$.
]

#example("Uncountable sets")[
  + The real numbers $RR$.
  + The real numbers in the interval $[0,1]$.
]

#remark[Bijection][
  If a set is countably infinite, then it has a bijection with $ZZ$. This means
  every set with cardinality $aleph_0$ has a bijection to $ZZ$.
]