Content
- 1 Event
- 1.1 Randomized experiment
- 1.2 Sample space
- 1.3 Point sample
- 1.4 Random event
- 1.5 Occurrence of an event
- 1.6 Event insurance
- 1.7 Event impossible
- 1.8 Event otherwise
- 1.9 Union of events
- 1.10 Intersection of events
- 1.11 Difference event
- 1.12 Symmetric difference of events
- 1.13 Laws of Morgan
- 1.14 Event incompatible
- 1.15 Set denumerable
- 1.16 Set accounting
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
- \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 - \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 - 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) - 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^cWith 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^cWith 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