Random Variable - Probability

8/3/2019 Random Variable - Probability

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By

Sreekanth MP110005EE

PhD Scholar

Dept of EE,

National Institute of Technology,

Calicut, India

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Almost Completion (or Settling)of now seen

number System (The Decimal) evolved

across the globe around the beginning of AD.

Of course with the inclusion of Zero from India.

Source: History of Mathematics, BBC World

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Exact existence: Not Known, Source: [email protected]



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Humans are trying to solve their mysterious Real Lifeequations with Real Line..!!.

Examples: Height

Weight

Age

Marks

Salary Bank Balance……etc..!!

Random Variables

Who ever solves the arbitrary - unwritten n-dimensional curve fitting problem with above variables are accepted as Winner…in that century itself..!.Some others…many centuries later…!! [email protected]



Definition : A random variable over a sample space is a function that maps every sample point (i.e. outcome)to a real number.(That is the Domain is S and the Range is some subsetof the real line).

A mapping X ( ) from S to the real line.

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Size of Spanner.•

Shoe Size

Electrical Wire Gauge

Seat Number in Train.

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"A gambler's dispute in 1654 led tothe creation of a mathematicaltheory of probability by twofamous French mathematicians,Blaise Pascal and Pierre de Fermat.

The game consisted in throwing apair of dice 24 times; the problem

was to decide whether or not to bet

even money on the occurrence of atleast one "double six" during the 24throws.

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Probability is a Mathematical Model to help us study physical systems in an average sense.

(Prob & RP with applications to Signal Processing, Stark & Woods, Pearson Education).

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Two of the most popular definitions are:

The relative frequency definition

The classical definition

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Suppose that a random experiment is repeated n times. If the event

A occurs nA times, then the probability of A, denoted by P(A) , isdefined as:

(nA /n) represents the fraction of occurrence of A in n trials.

For small values of n , it is likely that (nA /n) will fluctuate quite

badly.

But as n becomes larger and larger, we expect, (nA /n) to tend to a

definite limiting value.

Eg: Difference between Tossing a Coin 6 times and 100 times.

(We expect (nA /n) to converge to 0.5. )

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Experiments by:Number of

ThrowsNumber of Heads

RelativeFrequency of

Heads

Person 1 4040 2048 0.5069

Person 2 12000 6019 0.5016

Person 3 24000 12012 0.5005

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In tossing a fair coin, Let X = No. of trials until the firsthead appears. (Let the events are independent):

Then P(X=1) = P(H) = ½ . P(X=2) = P(TH) = ½ x ½ = ¼

P(X=3) = P(TTH) = ½ x ½ x 1/2 = 1/8. So P(X=n) = (1/2)n , for n = 1,2,3,…n. So it is sure that we will get a Head at nth trial.

That is the individual probabilities converge to 1, as n tends toa large value. i.e

Using Geometric Series the individual probabilitiesconverges to

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In the classical approach, the probability of the event A isfound without experimentation.

This is done by counting the total number N of the

possible outcomes of the experiment. If NA of those

outcomes are favorable to the occurrence of the event A,

then

where it is assumed that all outcomes are equally likely!

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Real Line

SE

0 1Probability

X(.) X(.)

P(.) P(.)

0 ≤ P(E) ≤ 1 [email protected]



Answer is NO, Because Mathematical modeling orautomated prediction methods are based on somereferences or some mappings like Random Variables.

We cannot manipulate with things, space or timeindependently.

That Gives Importance of Number system in our life.

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Random Variable - Probability

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