Event

Event

An event or set of events is each of the possible results of a random experiment

Randomized experiment

It is that which under similar conditions gives us different results

Examples of experiments random

  • Throw a coin and count the number of faces or crosses
  • Extract a card from a deck
  • Calculate the lifetime of a light bulb
  • Measure the temperature of a processor after an hour of work
  • Calculate the number of calls sent or received by a phone line after an hour

Sample space

The set is composed by all the possible outcomes associated with the randomized experiment

It is the entire set

It is represented with \Omega

Example of sample space

In the experiment of throwing a coin 3 times and counting the number of faces

The sample space will be \Omega=\{0,1,2,3\} for the number of faces obtained

Point sample

Single result obtained from a sample space

It is represented with \omega

Being A a set

And is defined p(\omega)=\{A|A\subseteq\Omega\}

Example of point sampling

In the experiment of throwing a coin 3 times and counting the number of faces

If after throwing the coin 3 times we have counted 2 faces, then the sample point is p(3)=2

Random event

It is a set of points sampling

It is represented with A

It is denoted with capital letters (A_i)_{i\in I} family (finite or infinite)

And is defined (A_i)_{i\in I} \in p(\Omega)

Example of a random event

In the experiment of throwing a coin 3 times and counting the number of faces

Let's repeat the experiment 5 times to get a random event, if after tossing the coin 3 times we have counted:

  • 2 faces, then sample point 1 is p(3_1)=2
  • 0 faces, then sample point 2 is p(3_2)=0
  • 3 faces, then sample point 3 is p(3_3)=2
  • 2 faces, then sample point 4 is p(3_4)=2
  • 1 face, then sample point 5 is p(3_5)=1

The random event is A=\{2,0,2,2,1\}

Occurrence of an event

We say that has occurred an event A if in a particular completion of the randomized experiment we obtain a point sample of P((A_i)_{i\in I})=\{A|A\subseteq\Omega\}

Example of the occurrence of an event

In the experiment of throwing a coin 3 times and counting the number of faces

We are going to repeat the experiment 5 times to obtain a random event

We are going to repeat the experiment 5 times to obtain a random event, if after flipping the coin 3 times we have obtained:

  • 2 faces, then the occurrence of the event is P(3_1)=2
  • 0 faces, then the occurrence of the event is P(3_2)=0
  • 2 faces, then the occurrence of the event is P(3_3)=2
  • 2 faces, then the occurrence of the event is P(3_4)=2
  • 1 face, then the occurrence of the event is P(3_5)=1

Event insurance

It is the one which happens always

It is represented with \Omega

Being A a set

It is denoted
p(\omega)=\{A|A\subseteq\Omega\}=\Omega
\Omega=\{x, x\in\Omega\}\not =\{\{x\},x\in\Omega\}\subseteq p(\Omega)

Example of event insurance

In the experiment of throwing a coin 3 times and counting the number of faces

Getting a number of faces (including 0) is a safe event because we can always count the number of faces (even if none comes out, because we've included 0)

The event sure, then it is
\omega=\{"get a number of faces"\}
p(\omega)=\Omega

Event impossible

It is the one that never happens

It is represented with \emptyset

Being A a set

It is denoted
p(\omega)=\{A|A\subseteq\Omega\}=\emptyset
p(\emptyset)=1

Example of event impossible

In the experiment of throwing a coin 3 times and counting the number of faces

Getting the color red, it is an impossible event because in the experiment we are taking into account the number of faces obtained, we are not taking into account the color of the die

The event sure, then it is
\omega=\{"get the color red"\}
p(\omega)=\emptyset

Event otherwise

We will call event otherwise A, to the event that occurs when it does not occur A

It is represented with A^c

It is denoted A^c=\Omega\backslash A

Example of an event contrary

In the experiment of tossing a coin and count the number of faces

Getting cross instead of face, is the opposite event because we're taking into account the number of faces, not crosses

If A=\{"number of faces obtained"\} then the event otherwise it is A^c=\{"number of crosses obtained"\}

Union of events

We will call event binding A and B, to the event that occurs or A or B or two

It is represented with A\cup B

Being A a set

It is denoted \underset{i\in I}{\bigcup} A_i\in p(\Omega)

Example of a union of events

In the experiment of tossing a coin and count the number of faces or crosses

Being
A=\{"number of faces obtained"\}=\{3,4\}
B=\{"number of crosses obtained"\}=\{2,4,6\}
A\cup B=\{"number of faces or crosses obtained"\}=\{2,3,4,6\}

Intersection of events

We will call the intersection of events A and B, to the event that occurs when it occurs A and B

It is represented with A\cap B

Being A a set

It is denoted \underset{i\in I}{\bigcap} A_i\in p(\Omega)

Example of intersection of events

In the experiment of tossing a coin and count the number of faces or crosses

Being
A=\{"number of faces obtained"\}=\{3,4\}
B=\{"number of crosses obtained"\}=\{2,4,6\}
A\cap B=\{"even number of faces and crosses obtained"\}=\{4\}

Difference event

We will call a difference event A and B, to the event that occurs when it occurs A or B but not both at the same time

It is represented with A \backslash B = A - B

It is denoted A - B = A - A \cap B = A \cap B^c

Example of difference of events

In the experiment of tossing a coin and count the number of faces or crosses

Being
A=\{"number of faces obtained"\}=\{3,4\}
B=\{"number of crosses obtained"\}=\{2,4,6\}
A-B=\{"odd number of faces or crosses obtained but not both at once"\}=A - A\cap B=\{3,4\}-\{4\}=\{3\}

Symmetric difference of events

We will call symmetric difference of events A and B, to the event of all events that occurs when it occurs A\cup B but not A\cap B

It is represented with A \triangle B

It is denoted A \triangle B = (A \cup B) - (A \cap B)

Example of symmetric difference of events

In the experiment of tossing a coin and count the number of faces or crosses

Being
A=\{"number of faces obtained"\}=\{3,4\}
B=\{"number of crosses obtained"\}=\{2,4,6\}
A\triangle B=\{"even number of faces or crosses obtained but not even number of faces and crosses"\}=(A \cup B) - (A \cap B)=\{2,3,4,6\}-\{4\}=\{2,3,6\}

Laws of Morgan

Laws proposed by Augustus De Morgan (1806-1871), an Indian-born British mathematician and logician, which set out the following fundamental principles of the algebra of logic:

  • The negation of the conjunction is equivalent to the disjunction of negations

  • The negation of the disjunction is equivalent to the conjunction of the negations

The following definitions of Morgan's laws can be used within the statistics:

Being A, B and C sets

  1. \left(A\cup B\right)^c = A^c\cap B^c
    whose generalized form is
    \left(\underset{i\in I}{\bigcup} A_i\right)^c = \underset{i\in I}{\bigcap} \left(A_i\right)^c
  2. \left(A\cap B\right)^c = A^c\cup B^c
    whose generalized form is
    \left(\underset{i\in I}{\bigcap} A_i\right)^c = \underset{i\in I}{\bigcup} \left(A_i\right)^c
  3. A\cap\left(B\cup C\right) = \left(A\cap B\right)\cup\left(A\cap C\right)
    whose generalized form is
    \underset{j\in I}{\bigcap}\left(\underset{i\in I}{\bigcup} A_i\right) = \underset{i j\in I}{\bigcup}\left(\underset{j\in I}{\bigcap} A_{i j, j}\right)
  4. A\cup\left(B\cap C\right) = \left(A\cup B\right)\cap\left(A\cup C\right)
    whose generalized form is
    \underset{j\in I}{\bigcup}\left(\underset{i\in I}{\bigcap} A_i\right) = \underset{i j\in I}{\bigcap}\left(\underset{j\in I}{\bigcup} A_{i j, j}\right)

Demonstration 1

We want to show that \left(A\cup B\right)^c = A^c\cap B^c

\omega\in\left(A\cup B\right)^c \Rightarrow \omega \not \in A\cup B \Rightarrow \begin{cases} \omega \not \in A \\ \omega \not \in B \end{cases} \Rightarrow \begin{cases} \omega \in A^c \\ \omega \in B^c \end{cases} \Rightarrow \omega \in A^c\cap B^c

With what we got to what we wanted, being tested

Demonstration 2

We want to show that \left(A\cap B\right)^c = A^c\cup B^c

\omega\in\left(A\cap B\right)^c \Rightarrow \omega \not \in A\cap B \Rightarrow \begin{cases} \omega \not \in A \\ \omega \not \in B \end{cases} \Rightarrow \begin{cases} \omega \in A^c \\ \omega \in B^c \end{cases} \Rightarrow \omega \in A^c\cup B^c

With what we got to what we wanted, being tested

Event incompatible

We will say that A and B these are events that are incompatible if they can not occur never at the same time

It is denoted
A \cap B = \emptyset
A \cap A^c = \emptyset

A family \left(A_i\right)_{i\in I} sets 2 to 2 disjoints (or mutually exclusive) if A_i\cup A_j = \emptyset when i\not = j

If a family \left(A_i\right)_{i\in I} is mutually exclusive, we'll denote it \underset{i\in I}{\sqcup}A_i := \underset{i\in I}{\cup}A_i

We say that a family \left(A_i\right)_{i\in I} is exhaustive if A_i\cap A_j = \Omega

Set denumerable

A set is said to denumerable if it is biyectivo with \mathbb{N}

Set accounting

A set is said to countable if a is denumerable or finite