MCom I Semester Statistical Analysis Theoretical Frequency Distribution Study Material notes

MCom I Semester Statistical Analysis Theoretical Frequency Distribution Study Material notes: Utility and Important  theoretical Frequency Distribution Binomial Distribution Types o Theoretical Frequency Distributions Conditions Assumptions of Binomial Distribution General Rules for Binomial Distributions Illustration based on Binomial Distributions Comparison of Actual and Expected Frequencies Poisson Distribution Conditions for Use of Poisson Distribution Assumption do Poisson Distrutbutin Characteristics of Poisson Distribution :

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Theoretical Frequency Distribution

Broadly speaking, the frequency distributions are of two types :

(1) Empirical or Observed Frequency Distribution, and

(2) Theoretical or Expected Frequency Distribution.

The observed frequency distribution is constructed on the basis of actual data or experimentation. For example, series relating to height measurements or marks obtained by students or wages of labourers are all obtained by observation on measurement of data. We may study the monthly wages of 500 labourers of a factory and classify the data in the form of a frequency distribution as follows:

Monthly wages (Rs.): 0-500 500-1.000 1.000-1,500 1,500-2,000 2,000-2,500

No. of Laborers :          80          120         50                  100                150

As distinguished from this type of distribution which is based on actual observations, it is possible to deduce mathematically what the frequency distributions of certain populations should be. Such distributions as are expected on the basis of previous experience or theoretical considerations are known as “theoretical distributions” or “expected distributions”. For example, if a coin is tossed we expect that as n increases we shall get close to 50% heads and 50% tails. If a coin is tossed 200 times, then our expectation will be 100 heads and 100 tails. Thus, it can be precisely concluded that Theoretical Distributions are not obtained by actual observations but are mathematically deduced on certain assumptions.

Such distributions which are not obtained by actual observations or experiments but are mathematically deduced on certain assumptions are called Theoretical Frequency Distribution”:

Utility and Importance of Theoretical Frequency Distribution

Theoretical frequency distribution is of great help in understanding and analyzing a large number of problems of practical life. They help in taking decisions in many situations. The theoretical distributions are useful in the following cases :

(1) It helps in finding out the nature of distribution under certain hypothesis or assumptions.

(2) This helps in forecasting.

(3) The theoretical frequency distribution helps in taking decisions on the face of uncertainty.

(4) Helpful in the comparison of actual and expected frequencies.

(5) Substitutes for actual distributions when to obtain the actual frequencies is costly or cannot be obtained at all.

(6) Many of the business and other problems can be solved on the basis of theoretical distributions.

Types of Theoretical Frequency Distribution

The various theoretical frequency distributions are as follows:

(1) Binomial Distribution

(2) Negative Binomial Distribution

(3) Poisson Distribution

(4) Multinomial Distribution

(5) Hypergeometric Distribution

(6) Normal Distribution

Among these, the first five distributions are of discrete type and the last one of continuous type. Binomial, Poisson and Normal three distributions are widely used in practice and hence shall be discussed in detail.

Binomial Distribution

The binomial distribution is also known as the ‘Bernoulli distribution after the Swiss mathematician James Bernoulli (1654–1705) who discovered it in 1700 and was first published in 1713, eight years after his death. The binomial distribution is a probability distribution expressing the probability of one set of dichotomous alternatives, i.e., success or failure. The probability of occurrence of an event is p and its non-occurrence is q.

Binomial Distribution or Bernoulli Distribution is the most fundamental and important discrete probability distribution in statistics and is defined by the probability Function:

Where N stands for the number of trials and n for the number of independent events.

Conditions/Assumptions of Binomial Distribution

(1) The number of trials of ‘n’ is finite and fixed.

(2) Trials are repeated under identical conditions.

(3) There is only two possible outcomes success (p) and failure (q) for each trial.

(4) A trial must be independent of each other.

(5) The probability of each trial remains constant and does not change from trial to trial.

General Rules for Binomial Distribution

(1) Number of Terms: The number of terms in a binomial expansion would always be n + 1. For example, there will be 5 terms in a binomial expansion (q + p) and there will be 8 terms in a binomial expansion of (+p.

(2) Order of Exponents : In a binomial expansion of (q + p)”. the exponents of q are n, (n-1), (1-2),…,1,0, respectively, and the exponents of pare 0.1,2,….(n-1), n, respectively (note:po = 1:0° = 1). It means in each succeeding term the power of q is reduced by one and the power of p is increased by one.

(3) Sum of Exponents: The exponents of p and q, for any single term, when added together, always sum to n.

(4) Numerical Coefficient: The coefficients for the n + 1 terms of the distribution are always symmetrical ascending to the middle of the series and then descending when n is odd number, n + 1 is even and the coefficients of the two central terms are identical.

The various terms of the numerical coefficients can be obtained either on the basis of laws of combination or from the Pascal’s triangle.

For n trials the binomial probability distribution consists of (n + 1) terms, the successive binomial coefficient being “CO, “C1,C2,…,”C,-1, “C..

To find the terms of the expansion, we use the expansion of (q + p)”. Since “Co = “C, = 1, the first and last coefficient will always be one. The binomial coefficient will be symmetric form. The values of the binomial coefficient for different values of n can be obtained easily from Pascal’s triangle given below:

Pascal’s Triangle

It can be easily seen that taking the first and last terms as 1. each term in the above table can be obtained by adding the two terms on either side of it in the preceding line (i.e., the line above it). As pointed out earlier it can be easily verified that the binomial coefficients are symmetric and the sum of the coefficients is 2″.

(5) The Order of p and q: Binomial distribution can be written in two forms viz., (q + p)” or (P +9)”. If the form of ( + )” is adopted, then the number of successes will be written in descending order. On the other hand, if the number

Conditions for Use of Poisson Distribution

Poisson distribution can be used mainly in the following condition

(1) n, the number of trials is indefinitely large, i.e., n o (11) p. the constant probability of success for each trial is indefinitely s a

1. p -0. (iii) np = m, (say), is finite.

Assumptions of Poisson Distribution

Poisson distribution is used under the following conditions:

(1) The variable is discrete.

(2) The event can only be either a success or a failure.

(3) The number of trials ‘n’ is finite and large.

(4) P (the probability of occurrence of an event) is so small that ‘q’ is almost equal to unity.

(5) The probability of an occurrence of the event is the same for any two intervals of equal length.

(6) Statistical independence is assumed, i.e. the occurrence or non-occurrence of the event in any interval is independent of the occurrence or non-occurrence in any other interval.

Characteristics of Poisson Distribution

The Poisson distribution possesses the following characteristics :

(1) Discrete Distribution: Like binomial distribution, Poisson distribution is also a discrete probability distribution, i.e., it is concerned with occurrences that can take integral values 0, 1, 2, ….

(2) Values of p and q: It is applied in situations where the probability of the success of an event (p) is very small and that of failure (q) is very high almost equal to 1, and n is also very large.

(3) Form of Distribution: The Poisson distribution is a positively skewed distribution, i.e., to the left. With the increase in the value of m, the distribution shifts to right and the skewness is reduced.

(4) Main Parameter : If we know m, all the probabilities of the Poisson distribution can be obtained. m is, therefore, called the parameter of the Poisson distribution.

(5) Basic Assumption: The basic assumption of Poisson distribution is that ‘m’ remains constant.

(6) Constants: The Poisson distribution is a discrete distribution with a parameter m. The various constants are:

Utility or Importance of Poisson Distribution

The conditions under which Poisson istribution is obtained as a limiting case of the Binomial distribution suggest that Poisson distribution can be used to explain the behavior of the discrete random variables where the probability of occurrence of the event is very small and the total number of possible cases is sufficiently large. As such Poisson distribution has found application in a variety of fields such as Queueing Theory (waiting time problems), Insurance, Physics, Biology, Business, Economics and Industry. Most of the Temporal Distributions (dealing with events which are supposed to occur in equal intervals of time, and the Spatial Distributions (dealing with events which are supposed to occur in intervals of equal length along a straight line) follow the Poisson Probability Law. We have given below some practical situations where Poisson distribution can be used :

(i) Number of telephone calls arriving at a telephone switch board in unit time (say, per minute).

(ii) Number of customers arriving at the super market: say per hour.

(iii) The number of defects per unit of manufactured product (This is done for the construction of control chart for c in Industrial Quality Control).

(iv) To count the number of radioactive disintegrations of a radioactive element per unit of time (Physics).

(v) To count the number of bacteria’s per unit (Biology).

(vi) The number of defective material say, pins, blades, etc., in a packing manufactured by a good concern.

(vii) The number of suicides reported in a particular day or the number of casualties (persons dying) due to a rare disease such as heart attack or cancer or snake bite in a year.

(viii) Number of accidents taking place per day on a busy road.

(ix) Number of typographical errors per page in a typed material or the number of printing mistakes per page in a book.

(i), (ii). (iv). (vii) and (viii) are examples of temporal distributions and the remaining are examples of spatial distributions.

Illustration 9.

The distribution of typing mistakes in the book is given below. Fit the Poisson Distribution.