## **BUSINESS STATISTICS NOTES**

B.COM 2ND AND 3RD SEM NEW SYLLABUS (CBCS PATTERN)

**Probability and Probability Distributions**

**Part B: Probability
Distribution - Binomial - Poisson - Normal**

**Part B: Probability Distribution - Binomial - Poisson - Normal**

**Meaning of Binomial
Distributions**

The binomial distribution also known as ‘Bernoulli Distribution’ is associated with the name of a Swiss mathematician James Bernoulli also known as Jacques or Jakob (1654-1705). Binomial distribution is a probability distribution expressing the probability of one set of dichotomous alternatives, i.e., success or failure.

This distribution has been used to describe a wide variety of processes in business and the social sciences as well as other areas. The type of process which gives rise to this distribution is usually referred to as Bernoulli trial or as a Bernoulli process. The mathematical model for a Bernoulli process is developed under a very specific set of assumption involving the concept of a series of experimental trials. These assumptions are:

1) An experiment is performed under the same conditions for a fixed number or trials, say, n.

2) In each trial, there are only two possible outcomes of the experiment.

3) The probability of a success denoted by p remains constant from trial to trial. The probability of a failure denoted by q is equal to (1 – p). If the probability of success is not the same in each trial, we will not have binomial distribution.

4) The trials are statistically independent, i.e., the outcomes of any trial or sequence of trials do not affect the outcomes of subsequent trials.

**Application
of Binomial distribution**

Binomial distribution is applicable when the trials are independent and each trial has just two outcomes success and failure. It is applied in coin tossing experiments, sampling inspection plan etc.

**Fitting a Binomial
Distribution**

When a binomial distribution is to be fitted to observe data, the following procedure is adopted:

1) Determine the values of p and q. If one of these values is known the other can be found out by the simple relationship p = (1 – q) and q = (1 – p). When p and p are equal the distribution is symmetrical, for p and q may be interchanged without altering the value of any term, and consequently terms equidistant from the two ends of the series are equal. If p and q are unequal, the distribution is skew. If p is less than ½ the distribution is positively skewed and when p is more than ½ the distribution is negatively skewed.

2) Expand
the binomial (q + p)^{n}. The power n is equal to one less than the
number of terms in the expanded binomial. Thus, when two coins are tossed (n =
2) there will be three terms in the binomial. Similarly, when four coins are
tossed (n = 4) there will be five terms, and so on.

3) Multiply each term of the expanded binomial by N (the total frequency), in order to obtain the expected frequency in each category.

**MEANING OF POISSON
DISTRIBUTION**

Poisson distribution is a discrete probability distribution and is very widely used in statistical work. It was developed by a French mathematician, Simeon Denis Poisson (1781-1840), in 1837. Poisson distribution may be expected in cases where the chance of any individual event being a success is small. The distribution is used to describe the behaviour of rare events such as the number of accidents on road, no. of printing mistake in a book, etc., and has been called “the law of improbable events”. In recent years the statisticians have had a renewed interest in the occurrence of comparatively rare events, such as serious floods, accidental release of radiation from a nuclear reactor, and the like.

The Poisson distribution is defined as:

**Role of the Poisson
distribution**

The Poisson distribution is used in practice in a wide variety of problems where there are infrequently occurring events with respect to time area, volume or similar units. Some practical situations in which Poisson distribution can be used are given below:

1. It is used in quality control statistics to count the number of defects of an item.

2. In biology to count the number of bacteria.

3. In physics to count the number of particles emitted from a radioactive substance.

4. In insurance problems to count the number of casualties

5. In waiting-time problems to count the number of incoming telephone calls or incoming customers.

6. Number of traffic arrivals such as trucks at terminals, aeroplanes in airports, ships at docks, and so forth.

7. In determining the number of deaths in a district in a given period, say, a year, by a rare disease.

8. The number of typographical errors per page in typed material, number of deaths as a result of road accidents, etc.

9. In problems dealing with the inspection of manufactured products with the probability that any one piece is defective is very small and the lots are very large.

10. To model the distribution of the number of persons joining a queue (a line) to receive a service or purchase of a product.

**MEANING OF NORMAL DISTRIBUTION**

The binomial and the Poisson distributions are the most useful theoretical distributions for discrete variables, i.e., they relate to the occurrence of distinct events. In order to have mathematical distribution suitable for dealing with quantities whose magnitude is continuously variable, a continuous distribution is needed. The normal distribution, also called the normal probability distribution, happens to be most useful theoretical distribution for continuous variables. Many statistical date concerning business and economic problems are displayed in the form of normal distribution.

The normal distribution was first described
by Abraham De Moivre (1667-1754) as the limiting form of the binomial model in
1733. Normal distribution was rediscovered by Gauss in 1809 and by Laplace in
1812. Both Gauss and Laplace were led to the distribution by their work on the
theory of errors of observations arising in physical measuring processes
particularly in astronomy. Throughout the 18^{th} and 19^{th}
centuries, various efforts were made to establish the normal model as the
underlying law ruling all continuous random variables, - thus the name normal.
These efforts failed because of the false premises. The normal model has,
nevertheless, become the most important probability model in statistical
analysis.

The normal distribution is an approximation to binomial distribution. Whether or not p is equal to q, the binomial distribution tends to the form of the continuous curve and when n becomes large at least for the material part of the range. As a matter of fact, the correspondence between the binomial and the curve is surprisingly close even for comparatively low values of n, provided that p and q are fairly near equality. The limiting frequency curve obtained as n becomes large is called the normal frequency curve or simply the normal curve.

The normal curve is represented in several
forms. The following is the basic from relating to the curve with mean

**Properties of the
Normal Distribution**

The following are the important properties of the normal curve and the normal distribution.

1) The normal curve is “bell-shaped” and symmetrical in its appearance. If the curves were folded along its vertical axis, the two halves would coincide. The number of cases below the mean in a normal distribution is equal to the number of cases above the mean, which makes the mean and median coincide. The height of the curve for a positive deviation of 3 units is the same as the height of the curve for negative deviation of 3 units.

2) The height of the normal curve is at its maximum at the mean. Hence the mean and mode of the normal distribution coincide. Thus for a normal distribution mean, median and mode are all equal.

3) There is one maximum point of the normal curve which occurs at the mean. The height of the curve declines as we go in either direction from the mean. The curve approaches nearer and nearer to the base but it never touches it, i.e., the curve is asymptotic to the base on either side. Hence its range is unlimited or infinite in both directions.

4) Since there is only one maximum point, the normal curve is Unimodal, i.e., it has only one mode.

5) The
points of inflexion, i.e., the points where the change in curvature occurs are mean
__+__ sd.

6) As distinguished from Binomial and Poisson distribution where the variable is discrete, the variable distributed according to the normal curve is a continuous one.

7) The first and third quartiles are equidistant from the median.

8) The
mean deviation is 4^{th} or more precisely 0.7979 of the standard
deviation.

9) The area under the normal curve distributed as follows: -

**Significance of the
Normal Distribution**

The normal distribution is mostly used for the following purposes:-

1. To approximate or “fit” a distribution of measurement under certain conditions.

2. To approximate the binomial distribution and other discrete or continuous probability distributions under suitable conditions.

3. To approximate the distribution of means and certain other quantities calculated from samples, especially large samples.

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