**Q.6.
Define probability distribution. Explain the salient features of Binomial,
Poisson and Normal**

**distribution.
(2008-09, 12-13)**

**Ans. Probability
Distribution:** According to Robert
Parket, probability distribution consists of mutually exclusive and exhaustive
compilation of all random events that process and the probability of each event
occurring. It is a mathematical model that represents the distribution of the
universe, the distribution shows the results we would obtain, if we take many
of the probability samples and compute the statistics for each sample. A table
listing all possible values that a random variable can take on together with
the associated probabilities is called a probability of distribution. The
probability distribution of X where X is the number of spots showing when a six
sided symmetrical die is rolled is as:

The sum of all probabilities in a probability distribution is 1 since, one of the values produced by a random experiment must occur, if the experiment is performed.

They serve as benchmarks against which to compare observed distributions which act as substitutes for actual distributions when the latter are costly to obtain. They provide decision-makers with a logical basis for making decisions and hence are useful in making various predictions on the basis of limited information or any theoretical consideration.

**Salient Features of Binomial, Poisson and Normal Distribution : Refer to Section-C, Q.7,8,9. **

**Q.7.
What is the binomial distribution? Explain the assumptions and properties of
binomial distribution. Or Write a note on binomial distribution.(2005-06) **

**Ans.
Binomial Distribution:** Binomial
distribution is associated with the name of Swiss mathematician James Bernoulli
(1654-1705). However, it was published in 1713, eight years after his death. It
is also known as Bernoulli’s distribution. Trial or process, the literal
meaning of the word ‘Binomial’ is two groups or dichotomous alternatives.
Hence, in this distribution, frequencies are divided on the basis of two possible
outcomes, which for the sake of convenience are called as success and failure.

**Definition
of Binomial Distribution**

Binomial distribution is a discrete frequency distribution, which is based on dichotomous alternatives, i.e. on the basis of probability of data of success (desired event) and failure. This distribution, in the form of probability density function, can be expressed as follows:

*P(x)= “Cxq”- *px or*

P (n)= “Cq-“p”

where, p= Probability of success,

q = Probability of failure or 1 – p,

n= Number of trials,

x and r = Number of successes in n trials.

**Conditions
for Applications or Assumptions of Binomial Distribution**

Binomial distribution can be applied only under following conditions:

**1
Finite Number of Trials:** In this
distribution, the number of trials should be finite and fixed, i.e. an
experiment is repeated under identical conditions for a fixed number of trials.

**2.
Mutually Exclusive Outcomes:** In each
trial, there must be only two possible outcomes of the event which are mutually
exclusive. For example, when we toss a coin, there are only two possible
outcomes: head or tail and one of them must happen.

**3.
Same Probability in Each Trial:** The
probability of the happening of an event (or success) denoted by pin each trial
remains constant. For example, the probability of a head or a tail is constant
for all unbaised throws of the coin.

**4.
All Trials Independent:** All trials
must be independent of each other, i.e. the result of any trial should not be
affected by the result of a previous or subsequent trial.

**5. Discrete Variable:** The variable should be discrete, i.e. failure or success should be in whole numbers.