Discrete Continuous Random (Stochastic or Chance) Variable

Discrete/Continuous Random Variable

A random variable may be • Discrete   • Continuous

Natural Numbers

The set of numbers 1, 2, 3, .... ∞ are called Natural Numbers or counting numbers. The set of natural numbers is named "N". [N = {1, 2, 3, .... ∞}]

The number of natural numbers is designated by À₀ [Read Aleph-null or Aleph-nought] {À - The first letter of the Hebrew alphabet } [The zero subscript is justified by the fact that no infinite set has a smaller cardinality than the set of natural numbers ]

Enumerable/Denumerable Set

A set which is finite or a set with a one-to-one correspondence with the natural numbers (or positive integers) is called an denumerable set.

The cardinality of the denumerable set is equal to that of the natural numbers.

A set is either countable, denumerable or uncountable.

• Countable

  A set is countable if its cardinality is either finite or equal to À₀.

  •   A null set is countable. P = {φ} ⇒ n(P) = 0

  •   A finite set is countable V = {a, e, i, o, u} ⇒ n(V) = 5

• Denumerable

  A set is denumerable if its cardinality is exactly À₀

  A Set of Integers is Denumerable

  •   Z = {−∞,....., −3, −2, −1, 0, 1, 2, 3, ..... ∞}

  The elements of the set may not seem to be countable, but they are

  Re-arranging them as below we can count them...

  •   Z = {0, 1, −1, 2, −2, 3, −3, ..... ∞ −∞}

  The number of elements in a set is infinite does not mean that they are not countable.

  Interlacing

  The method of alternating the members of two infinite sequences in order to make a single sequence is called interlacing.

  A Set of Rational Numbers is Denumerable

• Uncountable

  A set is uncountable if its cardinality is greater than À₀.

  A Set of Real Numbers is Uncountable

  The cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers.

  The number of real numbers between 0 and 1 is the same as the number of all real numbers.

Discrete/Continous Sample Space

A sample space may be either discrete or continous.

• Discrete Sample Space

  The sample space is discrete if it is countable i.e. if the number of elements in the set are finite or denumerable.

• Continous Sample Space

  The sample space is continous if it is not countable i.e. if the number of elements in the set are not denumerable.

Discrete Random Variable

A real valued function defined on a discrete sample space is called a discrete random variable.

Since it is defined on a discrete sample space its range would also be discrete. The set forming the range of the random variable has a countable number of values/elements.

The number of values that the variable representing an element in the range ("x" in the above examples) takes is always countable.

Eg:	1.	In the experiment of tossing 3 coins, If "S" represents the sample space, "s" a sample point. If "X" is a random variable such that X (s) = x, where "x" represents the number of heads, then "x' can take only four values either 0, 1, 2, or 3 ⇒ For "X" domain is "S" and Range = {0, 1, 2, 3} ⇒ "X" is a Discrete Random Variable "X' is a random variable as it associates each outcome of the experiment with a numerical value and it is a discrete random variable as its range is countable (all finite sets are countable).
	2.	In the experiment of rolling a die, If "S" represents the sample space, "s" a sample point. If "Y" is a random variable such that Y (s) = x, where "x" represents the number appearing on the die, then "x' can take only six values either 1, 2, 3, 4, 5 or 6 ⇒ For "Y" domain is "S" and Range = {1, 2, 3, 4, 5, 6}] ⇒ "Y" is a Discrete Random Variable "Y' is a random variable as it associates each outcome of the experiment with a numerical value and it is a discrete random variable as its range is countable (all finite sets are countable).
	3.	A person picking 3 balls from a bag can win Rs. 10 for each blue ball drawn and lose Rs. 5 for each red ball drawn. In the experiment of drawing 3 balls from a bag containing 5 red and 4 blue balls, where "R" represents a red ball and "B" represents a blue ball 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 respectively The sample space S = (BBB, BBR, BRB, RBB, RRB, RBR, BRR, RRR} {Where BBR ⇒ (B1, B2, R3), .....} Let "s" represent a sample point and "x" the amount that the person wins Let "X" be a random variable such that X (s) = x. The values of "x" as determined by the random experiment would be Balls Drawn Amount Won (x) Function 3 Blue + 30 [{3×10} − {0×5}] X(BBB) = + 30     (Or)     X : BBB → + 30 2 Blue 1 Red + 15 [{2×10} − {1×5}] X(RBB) = + 15 X : RBB → + 15 X(BRB) = + 15 X : BRB → + 15 X(BBR) = + 15 X : BBR → + 15 1 Blue 2 Red 0 [{1×10} − {2×5}] X(RBB) = 0 X : RRB → 0 X(RBR) = 0 X : RBR → 0 X(BRR) = 0 X : BRR → 0 3 Red −15 [{0×10} − {3×5}] X(RRR) = −15 X : RRB → −15 "x' can take only four values either −15, 0, +15 or +30 ⇒ For "X" domain is "S" and Range = {−15, 0, +15 or +30}] "X" is a Discrete Random Variable "X' is a random variable as it associates each outcome of the experiment with a numerical value and it is a discrete random variable as its range is countable (all finite sets are countable). This relationship can be summarised as Event [Balls drawn being]       BBB RBB BRB BBR RRB RBR RRB RRR "x" + 30 + 15 + 15 + 15 0 0 0 − 15

Number of real values within a range Hide/Show

Number of values that a variable may have within a particular range is infinite if the variable can carry any real value within that range. The number of real numbers between 0 and 10 is infinite, since between two given real numbers there always exists another real number. Eg: 1. Say 13 27 and 17 27 are two given real numbers. We can always locate another real number that lies between these two. The new number existing between the two given numbers can always be identified by taking a fraction with its numerator as the sum of the numerators of the given numbers and denominator as the the sum of the denominators of the given numbers. 13 + 17 27 = 30 27 = 5 9 [ 13 27 = 0.48, 17 27 = 0.63, 5 9 = 0.56, ] 5 9 is between 13 27 and 17 27 If we taken 13/27 and 5/9 and find a number between those two, it also lies between 13/27 and 17/27. In this way this process can be repeated infinite times to obtain infinite real values lying between 13/27 and 17/27. This should lead us to the conclusion that between two given real values, the number of values that can be assumed is infinite. This is important because whenever we are given that a particular variable can hold any real value or any real value within a certain range we should understand that the number of values that the variable may hold is infinite. Eg: 1. If a variable "x" can assume all possible real values, then the number of values that the variable may assume is infinite. 2. If a variable "x" can assume all possible real values between 0 and 5, then also the number of values that the variable may assume is infinite.

Continuous Random Variable

A Continuous random variable is a random variable which can take all real values in an interval. The range of a continous random variable is a set consisting of all real values within a range.

It would not be possible to enumerate/count the number of values that the variable representing an element in the range takes.

Eg:	1.	In the experiment of testing for the time taken to complete a task by workers If "S" represents the sample space, "s" a sample point. If "X" is a random variable such that X (s) = x, where "x" represents the height of a person, then "x' can take any value between 1 and 5 (assuming that the minimum time recorded is 1 minute and maximum is 5 minutes) [⇒ For "X" domain is "S" and Range = (0, 5)] ⇒ "X" is a Continuous Random Variable "X' is a random variable as it associates each outcome of the experiment with a numerical value and it is a continuous random variable as its range is uncountable
	2.	In the experiment of finding the heights of persons If "S" represents the sample space, "s" a sample point. If "X" is a random variable such that X (s) = x, where "x" represents the height of a person, then "x' can take any real value between 0 and 12 (say if the heights are expressed in feet and assuming that the maximum height recorded is 12 ft.) [⇒ For "X" domain is "S" and Range = {0, ..... , 12}] ⇒ "X" is a Continuous Random Variable "X' is a random variable as it associates each outcome of the experiment with a numerical value and it is a Continuous random variable as its range is uncountable.
	3.	In the experiment of testing for the % of cane juice in sugar cane If "S" represents the sample space, "s" a sample point. If "X" is a random variable such that X (s) = x, where "x" represents the % of juice in sugar cane, then "x' can take any real value between 0 and 100. [⇒ For "X" domain is "S" and Range = {0, ..... , 100}] ⇒ "X" is a Continuous Random Variable "X' is a random variable as it associates each outcome of the experiment with a numerical value and it is a Continuous random variable as its range is uncountable.