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.