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Random or Stochastic or Chance Variable

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Outcomes of an Experiment

An outcome is something that results. Therefore the outcomes of an experiment are the results of the experiment. They are also termed choices.

In measuring probabilities, we consider all the possible outcomes/choices of the experiment as well as the favourable outcomes/choices which are the outcomes/choices which are favorable to the occurrence of an event.
Eg: 1. In the experiment of tossing 3 coins, there are eight possible outcomes.

Where "H1, H2, H3" represents a head on the first, secon and third coins respectively and "T1, T2, T3" represents a tail on the first, second and third coins respectivley, the possible events can be represented as

•   (H1, H2, H3)

•   (H1, H2, T3) (Or) (H1, T2, H3) (Or) (T1, H2, H3)

•   (H1, T2, T3) (Or) (T1, H2, T3) (Or) (T1, T2, H3)

•   (T1, T2, T3)

2. In the experiment of rolling a die, there are six possible outcomes, the die showing up

• ONE   • TWO   • THREE   • FOUR   • FIVE   • SIX

3. In the experiment of drawing 3 balls from a bag containing 5 red and 4 blue balls, there are five possible outcomes.

Where "B1, B2, B3" represents the first, second and third balls being blue respectively and "R1, R2, R3" represents the first, second and third balls being red respectivley, the possible events can be represented as

• (B1, B2, B3)

• (B1, B2, R3) (Or) (B1, R2, B3) (Or) (R1, B2, B3)

• (B1, R2, R3) (Or) (R1, B2, R3) (Or) (R1, R2, B3)

• (R1, R2, R3)

As seen above, the outcomes of an experiment need not be numbers. However, at times we need to represent outcomes as numbers. That necessity is served by a random variable.

A random variable is a function that associates a numerical value with every outcome of an experiment.

Some Terms You may find useful Hide/Show

  • Set

    A set is a collection of objects. An object can be anything that is conceivable in the mind. Each object belonging to a set is called an "element" of the set. For a group of objects to form a set, it should be possible to verify whether a given object belongs or not belongs to the group.
  • Cardinal Number

    Numbers may be either cardinal or ordinal. A cardinal number lets us know "how many?" and an ordinal number lets us know "which one?"

    The cardinality of a set is the number of elements/members it contains. Cardinal number is a number that denotes the number of elements in a set. It is represented as n(Set Name) = No. of elements.

    Where S = {1, 2, 3, 4, 5, 6} ⇒ n(S) = 6

  • Finite Set

    A set with a finite number of elements in it.
    Eg: 1. Where S = {1, 2, 3, 4, 5, 6} ⇒ n(S) = 6. "S" is a finite set
    2. Where A = {x/x N and x (1, 100)} ⇒ n(A) = 100. "N" is a finite set

    ["N" represents the set of natural numbers]

  • Infinite Set

    A set with infinite number of elements in it.
    Eg: 1. Where S = {1, 2, 3, .... ∞} ⇒ n(S) = ∞. "S" is an infinite set
    2. Where B = {x/x Q ⇒ n(B) = ∞. "B" is a infinite set

    [Since "Q" represents the set of Rational numbers and they are infinite]

  • Ordered Pair

    Any two objects taken together would form an ordered pair. They are represented as (First Object, Second Object).

    The first object is called the first coordinate or first member and the second object is called the second coordinate or second member of the ordered pair.

    When being represented as a set, an ordered pair is represented as { {First Object}, {First Object, Second Object} }
    Eg: 1. Where "p" and "q" are any two objects,
    • (p, q)forms an ordered pair where

      "p" is the 1st coordinate/member and "q" is the 2nd coordinate/member.

      In a set form (p, q) is represented as { {p}, {p, q}}

    • (q, p) forms an ordered pair where

      "q" is the 1st coordinate/member and "p" is the 2nd coordinate/member.

    • In a set form (q, p) is represented as { {q}, {q, p}}

    In an ordered pair the order of the objects/elments is important. (p, q) is not the same as (q, p)

  • Cartesian Product of Sets

    A Cartesian product of two sets is the set of all possible ordered pairs taking the first member from the a set and the second member from the other set. It is represented using the names of the two sets with a "X" (read cross) placed between them.

    The cartesian product of the two sets "A" and "B", is the set of all possible ordered pairs (a, b), where a A and b B.

    ⇒ [A X B = { (a, b)/ a A and b B}
    Eg: 1. Where A = {a, b, c} and B = (4, 5, 6)
    A X B = {(a, 4), (a, 5), (a, 6), (b, 4), (b, 5), (b, 6) , (c, 4), (c, 5), (c, 6)}
    B X A = {(4, a), (4, b), (4, c), (5, a), (5, b), (5, c), (6, a), (6, b), (6, c),}

    In a cross product set, the order of the sets is important. "A X B" is not the same as "B X A"

  • Relation

    A relation is a set of ordered pairs. It is a subset of the set of the cross product between two sets. The first and second members of the ordered pairs within the relation satisfy a common criteria called the rule of the relation.

    Where "A" and "B" are two sets, A X B the Cross Product Set, then the set "R" is a relation from "A to B" ⇒ R ⊆ A X B

    The relation is represented as (a, b) R (Or) a R b [Read as "a" is in the relation "R" to "b"]
    Eg: Where A = {1, 2, 3} and B = (4, 5, 6)

    A X B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6) , (3, 4), (3, 5), (3, 6)}

    If a A and b B

    1. If R = {(2, 4), (2, 6)}, then "R" is a relation from "A" to "B"

    [Since R ⊆ A X B and all the elements of "R" satisfy a common rule "a" is even ⇔ "b" is even]

    2. If M = {(1, 4), (1, 6), (3, 4), (3, 6)}, then "M" is a relation from "A" to "B"

    Since M ⊆ A X B and all the elements of "M" satisfy a common criteria/rule "a" is odd ⇔ "b" is even

  • Domain and Range of a Relation

    The set of first coordinates of the elements of a relation form its domain and the set of second coordinates of the relation form its range.
    Eg: Where A = {1, 2, 3} and B = (4, 5, 6)

    A X B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6) , (3, 4), (3, 5), (3, 6)}

    If a A and b B

    1. If R = {(2, 4), (2, 6)}, then "R" is a relation from "A" to "B"

    Domain = {2} and Range = {4, 6}

    2. If M = {(1, 4), (1, 6), (3, 4), (3, 6)}, then "M" is a relation from "A" to "B"

    Domain = {1, 3} and Range = {4, 6}

  • Function/Mapping

    A function is a relation. A function is a relation which associates each element of the domain to a single element of the range. The terms Mapping and function are synonymously used.

    For two sets "A" and "B", if every element in set "A" is associated with a single element in set "B" we say that "f" is a function from "A to B" or "f" maps "A to B" or "f" is a mapping from "A to B".

    Thus, for the function "f'", "A" is the "Domain" and "B" is the "Range".

    [Alternatively :: A function is a definition of the relationship between two variables. The two variables are so related that for every value of the first variable there is a unique (only one) value of the second.]

    It may be noted that an element in "Set B" may be associated to two or more elements in "Set A", but an element in "Set A" should be associated with only one element in "Set B".

    A function is generally named (indicated by the symbol) "f" or any other name like "Cube". Where we do not intend to use a name for the function, it is expressed in terms of the two variables involved in the relation.
    Eg: 1. f (x) = x² indicates a function. It relates x → x².

    For each value of x, there is a single value x² associated with it.

    This may also be written as

    • y = x²

      This is done where we do not intend to use a name for the function. In such case the two variables involved are identifiable and the function representing the relationship between the two variables is written as an equation between the two variables.

    • square (x) = x²

      Here "square" is the name of the function [being used in place of "f"]

      So, what would the words Log (in Log m), Tan (in Tan θ),.... etc., be. They are also names of functions. Logarithmic Function, Trigonometric Function.

      Each element in the first set (values of x) has only one element in the second set related to it. However an element in the second set has two elements in the first set associated with it. [f(+3) = 9 ⇒ x=+3, y = 9; F(−3) = 9 ⇒ x=−3, y = 9;]

    A function need not involve numbers.

    Eg: 1. Capital (Country) = City
    "Capital" is a function that maps a country to its capital city.
    [Capital : Country → City] Eg: Capital (Japan) = Tokyo
    2. X (Game) = Field Shape.

    "X" is a function that maps a Game to the shape of the field/board on which we play it

    We say "X" is a function from "Game" to "Field Shapes" (Or) X : Game → Field Shapes.

    Each Game (element of the first set) is associated with only one field shape (element of the second set). However, a field shape (element in the second set) may be associated with two or more games (elements in the first set).
    Eg: X (Cricket) = Oval, X (Football) = Rectangle,

Random or Stochastic or Chance Variable

  • Sample Space

    The set of all possible elementary events in a random experiment.
  • Sample Points

    The elements of the Sample Space i.e. the elementary events in the experiment
  • Stochastic

    A variable quantity that is random

A random variable is a function/mapping which connects each element of a sample space to a particular numerical value. A random variable is also called a stochastic or chance variable.
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Let "S" and "s" represent the sample space and sample points of a random experiment.

The function "X" which assigns to each "s" ( S) a real number "x" is called a random variable and is written as X(s) = x (Or) X : s → x. The domain of "X" is the "set S" and its range is a Set of real numbers carried by "x", defined by Range = {x/x = X(s), s S}

Every outcome of a random experiment is associated with a single numerical value by the random variable.

Eg: 1. In the experiment of tossing a die

The sample space S = (ONE, TWO, THREE, FOUR, FIVE, SIX}

If "s" represents the sample points and "x" the number appearing on the die

Then,

The random variable "X" representing the relationship between "s" and "x" is

X(s) = x (Or) X : s → x

The value of "x" as determined by the outcome of the experiment would be
Dice Shows up Value of "x" Function
ONE 1 X(ONE) = 1     (Or)     X : ONE → 1
TWO 2 X(TWO) = 2 X : TWO → 2
THREE 3 X(THREE) = 3 X : THREE → 3
FOUR 4 X(FOUR) = 4 X : FOUR → 4
FIVE 5 X(FIVE) = 5 X : FIVE → 5
SIX 6 X(SIX) = 6 X : SIX → 6

"X" is a function with domain "S" and Range = {1, 2, 3, 4, 5, 6}.

This relationship can be summarised as
Event [Die Showing Up]       ONE TWO THREE FOUR FIVE SIX
"x" 1 2 3 4 5 6

2.

In the experiment of tossing 3 coins, where "H" represents a head and "T" a tail on the coin,

The sample space S = (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Where HHT ⇒ (H1, H2, T3), .....}

If "s' represents the sample points and "x" the number of heads obtained

Then,

The random variable "X" representing the relationship between "s" and "x" is

X(s) = x (Or) X : s → x

The value of "x" as determined by the outcome of the experiment would be
No. of Heads Shown up Value of "x" Function
ZERO 0 X(TTT) = 0     (Or)     X : TTT → 0
ONE 1 X(HTT) = 1 X : HTT → 1
X(THT) = 1 X : THT → 1
X(TTH) = 1 X : TTH → 1
TWO 2 X(HHT) = 2 X : HHT → 2
X(HTH) = 2 X : HTT → 2
X(THH) = 2 X : THH → 2
THREE 3 X(HHH) = 3 X : HHH → 3

"X" is a function with domain "S" and Range = {0, 1, 2, 3}.

This relationship can be summarised as
Event [Coins Showing Up]       HHH THH HTH HHT TTH THT TTH TTT
"x" 3 2 2 2 1 1 1 0

Some consider the expression random variable a misnomer, as a random variable is not a variable but rather a function that maps events to numbers.

A random variable cannot be assigned a value; It does not describe the actual outcome of a particular experiment, but rather describes the possible, outcomes in terms of real numbers.

Author Credit : The Edifier ...Continued Page 2

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