Theoretical Distributions

8/2/2019 Theoretical Distributions

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Theoretical Distributi


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Order is heavens first


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Theoretical Distributions of a Random variable

An empirical listing of outcomes and their observfrequencies.

A subjective listing of outcomes associated with thor contrived probabilities representing the degree othe decision maker as to the likelihood of the possib

Theoretical listing of outcomes and probabilitibe obtained from a mathematical model representiphenomenon of interest.

(Keywords - Outcomes, probabilities)


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Theoretical Distributions of a Random variableTheoretical listing of outcomes and probabilities obtained from a mathematical model representing phenomenon of interest.Apart from observed frequency distributions whic

obtained by grouping data, it is also possible to demathematically what distributions of certain popushould be. Such distributions as are expected on previous experience or theoretical considerationsare known as Theoretical Distributions.

(Keywords - observed frequency distributions , prev


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Theoretical Distributions of a Random VariabExampleIf a coin is tossed we expect that as n increases we50% heads and 50% tails.On the basis of this expectation we can test whether the cnot.If a coin is tossed 100 times, we may get 40 heads &This is our observation, whereas our expectation is 50 %

tails.

The questions is whether this discrepancy is due to due t

fluctuations or is due to the fact that the coin is biased.The fact that probabilities for both heads and tails mean that we must always get 50% heads and 50%IT MEANS THAT IF EXPERIMENT IS CARRLARGE NUMBER OF TIMES WE WILL OAVERAGE GET CLOSE TO 50%HEADS/TAI


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Theoretical Distributions of a Random VariabA probability distribution for a discrete random varimutually exclusive listing of all possible numerical othat random variable such that a particular probabilioccurrence is associated with each outcome.

Face of Outcome Probability1 1/62 1/63 1/64 1/65 1/66 1/6

PROBABILITY DISTRIBUTION OF THE RESULTS OF ROLLING O

Siouinco(oexthsu

(e.g.Prob of getting 4 is 1/6; Prob.of getting even number =3/6 ,Prob.of getti


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Theoretical Distributions of a Random VariabPl. be noted, A RANDOM VARIABLE IS A NUMERICAL QUWHOSE VALUE IS DETERMINED BY THE OUTRANDOM (CHANCE) EXPERIMENT.When a random experiment is performed , theoutcomes of the experiment forms a set which is caSpace(S) of the experiment.

(Keywords-Random experiment , sample space)


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Theoretical Distributions of a Random VariableLet the random experiment be tossing of a coin 2

Here S={ (T,T),(T,H),(H,T),(H,H)},then the number of heads obtained in both the t(T,T) 0(T,H) 1(H,T) 1(H,H) 2The sample space can be written as S = {0,1,2}


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Theoretical Distributions of a Random VariableHere S={ (T,T),(T,H),(H,T),(H,H)},then the number of heads obtained in both the trial shall be:(T,T) 0(T,H) 1(H,T) 1(H,H) 2The sample space can be written as S = {0,1,2}

P (X=0) = P (T,T) = P(X=1) =P[(T,H),(H,T)]= P(X=2) = P[(H,H)] = Hence P (X)= + + = 1Such a function P(X) is called the probability furandom variable X.The probability distribution is the outcomeprobabilities taken by this function of the random vari

(Keywordsprobability function, probability distr


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Theoretical Distributions of a Random VariabA Random variable can be Discrete or Continuous.A Random Variable is said the be Discrete if the sedefined by it over the sample space is finite , and itsprobability function P(X) is called as Probability Mand its distribution is called Discrete Probability DisA Random Variable is said to be Continuous if it caany (real) value in an interval, and its probability fuis called Probability Density Function and its distrcalled Continuous Probability Distribution.


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Theoretical Distributions of a Random VariabAmong the Theoretical or expected frequency dthe following six are more popular :1.Binomial Distribution2.Mutinomial Distribution3.Negative Binomial Distribution4.Poisson Distribution5.Hypergeometric Distribution &6.Normal Distribution.


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Binomial & Poisson Distribution.ppt
http://binomial%20%26%20poisson%20distribution.ppt/http://binomial%20%26%20poisson%20distribution.ppt/


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Normal Distribut


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The Normal Distribution also called the Normal Probability Distribution, hathe most useful theoretical distribution for continuous variables.

Normal Distribution is the cornerstone of modern statistics.The Normal Model has become the most important probability model in staThe normal distribution is an approximation to Binomial Distribution wheth

equal to q, the Binomial Distribution tends to be the form of the continuousn becomes large, at least for the material part of the range.The correspondence between Binomial and Normal curve is close even for

low values ofn, provided that p & q are fairly near equality.The normal frequency curve is represented in several forms.


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Normal Distribution which is also called the Normal Curve, is the mtheoretical distribution which describes the expected distribution of sand many other chances of occurrences.The normal curve is bell shaped and almost 99% of its values are wi 3 standard deviations from its mean.


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100 115857055 130 145 IQ

In this example a Standard Deviation for IQ equals 15.We can identify the proportion of the curve by measuring a scores distance (in this casefrom the mean (100)

Fig: Normal Distribution


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The Normal Distribution-(x - )

X= Values of the continuous random variable= Mean of the normal random variablee= mathematical constant approximated by 2.7183=Mathematical constant approximated by 3.1416(2 = 2.5066)

The following is the basic form relating to the curve with mean and standard

2

22eP(X) = 1

e


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The Equation of Normal Curve-x

The quantity N is equal to the maximum ordinate( yo) of the normal curve corresponding to the distriof stated total frequency N and stated standard devia

2

22e y = N

e

2

2


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GRAPH OF NORMAL DISTRIBUTIORemarks & Observations:1.The Normal Distribution can have different shapes depending on di & but there is one and only one normal distribution for any given for & .2.Normal Distribution is a limiting case of Binomial Distribution whena) n andb) neitherp nor q is very small.3.Normal Distribution is a limiting case of Poisson Distribution when i4.The mean of a normally distributed population lies at the centre of it5.The two tails of the normal probability distribution extend infinitely

touch the horizontal axis (which implies a positive probability for findinrandom variable within any range from minus infinity to plus infinity.)


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IMPORTANCE OF THE NORMAL DISTRIB1.The Normal Distribution has the remarkable property stated in the

Central Limit Theorem( CLT).

According to this theorem as the sample size n increases the distributof a random sample taken from practically any population approachedistribution (with mean & standard deviation /n).

Thus ,if samples of large size, n, are drawn from a population that is ndistributed ,nevertheless, the successive sample means will form them

distribution that is approximately normal.Hence, as the size of the sample is increased the sample means will tend to be distributed.The CLTapplies to the distribution of most other statistics such as Me

Deviation (but not range).CLTgives the Normal Distribution its central place in the theory of sammany important problems can be solved by this single pattern of samp


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PROPERTIES OF THE NORMAL DISTRIBUT1.The normal distribution is bell shaped and symmetrical in its appear

curves were folded along its vertical axis, the two halves would coincide2.The number of cases below the mean in a normal distribution, is equanumber of cases above the mean , which make mean & median coinc3.The height of the curve for a positive deviation of 3 units is the same a

the curve for negative deviation of 3 units.4.The height of the normal curve is at its maximum at the mean , hence

mode of the normal distribution coincide.Thus for a normal distribution mean , median & mode are all equal.


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PROPERTIES OF THE NORMAL DISTRIBUT5.There is one maximum point of the normal curve which occurs at theheight of the curve declines as we go in either direction from the meanapproaches nearer and nearer to the base but it never touches it i.e. theAsymptotic to the base on either side, hence its range is unlimited or indirections.6.Since there is only one maximum point, the normal curve is unimod

one mode.7.The points of inflexion i.e. the points where the change in curvature o8.In Binomial & Poisson distribution the variable is discrete whereas inDistribution the variable distributed is continuous9.The first and third quartiles are equidistant from the Median.


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PROPERTIES OF THE NORMAL DISTRIBU10.The mean deviation is 4th or more precisely 0.7979 of the standard11.The area under the normal curve distributed as follows:

Mean 1 covers 68.26% area (34.135% area will lie on either side of tMean 2 covers 95.45% areaMean 3 covers 99.73 % area.

Descriptive Statistics.ppt
http://descriptive%20statistics.ppt/http://descriptive%20statistics.ppt/


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AREA RELATIONSHIPDistance from the Mean Ordinate Percentage of Total Area

0.5 19.1461.0 34.1341.5 43.3191.96 47.5002.0 47.7252.5 49.379

2.5758 49.5003.0 49.865


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Distance frMean Ordi0.5 1.0 1.5 1.96 2.0 2.5 2.5758 3.0

AREA RELATIONSHIP Thus the two ordinates at distance 1.96 from the meanon either side would enclose 47.5 +47.5=95% of the total

area. The two ordinates at 2.5758 distance from the mean oneither side would enclose 49.5+49.5=99% of the total area. The area enclosed between ordinates at 3 distance fromthe mean on either side would be 49.865 +49.865 =99.73%

of the total area. The various hypothesis are tested either at 5% level or at 1%level(i.e. taking into account 95% & 99% of the total area of the normal curve )


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CONDITIONS FOR NORMALITY1.The causal forces must be numerous and of approximately equal we2.These forces must be the same over the universe from which the ob

drawn(although their incidence will vary from event to event).This is the condition of homogeneity

3.The forces affecting events must be independent of one another.4.The operations of causal forces must be such that deviation above th

mean are balanced as to magnitude and number by deviations belowThis is the condition of symmetry.


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CONSTANTS OF THE NORMAL DISTRI The mean of the Normal Distribution is X The standard deviation of the normal distribution is

2 =2 ; 3=0 and 4=341 or moment of coefficient of Skewness1 = 32 = 032

2 or moment of coefficient of Kurtosis2 = 4 = 34

22 4


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Areaunder the

Normal Curve


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The equation under the normal curve gives the ordinate of the curve correspany given value of x :

y = N

2

e22-x2

Although the researchers are usually are more interested in areas undecurve instead of its ordinate.The areas under the curve gives us the proportion of the cases falling bnumbers or the probability of getting a value between the two number


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As a researcher it is important to understand the meaning of the normastandard form.The equation of the normal curve depends on X and , and for its difX and we will obtain different curves (pl remember we calculate the ause z table).Since for different values, different tables will be required hence it was standardize the data (which is done through the use of one table)So now we can determine the normal curve areas regardless of X and the area under the normal curve having X =0 and =1.Such a Normal curve with 0 mean and unit Standard Deviatas the Standard normal curve.


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P(Z)= N2

e-x2

2

The standard normal probability curve is given by the equation- Z

A normal curve with mean X and standard deviation can be convertedstandard normal distribution by performing the change of the scale and ( as discussed above).In the original scale ( thex scale) the mean and the standard deviation arthe new scale ( thez-scale) they are 0 and 1.The formula that enables us to changex-scale toz-scale and vice versa is

z= X-X or x

wherex= (X- X)This transformation from X to z is named as z-transformation and has the e

X to units in terms of standard deviation

f(z)


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0

X-1 12 2 33

X -3 X -2 X - X +3+2+x-valuesz-values 68.27%

95.45%99.73%

Fig: The Standardized Normal Distribution

f(z)


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0

-1 12 2 33

-3 -2 -1 +3+2+1x-valuesz-values 68.27%

95.45%99.73%

Fig: The Standardized Normal Distribution


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Given a value of X, the corresponding value of z tells us how far away direction X is from its mean in term of its standard deviation .

Fig: Standardized Nor


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0 1123 2 3 Z

With the help of standardized normal distribution researchers can find the probPortion of the area under the standardized normal curve. All we have to do is trconvert the data from other observed normal distributions to the standardized nIn other words, the standardized normal distribution is extremely valuable becauor transform any normal variable , X into the standardized value Z


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Computing the standardized value, Z, of any measurement expressed iis simple:Subtract the mean from the value to be transformed and divide the stan(all expressed in original units).Here the population standard deviationthe formula: Z= X*- *(here X=normal ran

Standard Value = (Value to be transformed) (Mean)Standard Deviation

Where = hypothesized or expected value of the mean.

(source: Will

Linear Transformation of any Normal Variable into a Standardized Norm


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012 1 2

Z= X -

Sometimshrunkometimes the

scale is stretched


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Illustrations1.Find the area under the normal curve for z = 1.54Ans: From the table, the entry corresponding toz=1.54 is 0.4382 and this mea

shaded area in the following figure betweenz = 0 &z = 1.54

+1 +2 +3123 0

0.4382


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Theoretical Distributions of a Random Variab


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Face of Outcome Probability1 1/62 1/63 1/64 1/65 1/66 1/6


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Theoretical Distributions