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Binomial Distribution and Derivation of Binomial Distribution
Some discrete variable can take on many values, such as
1. Test score from 0 to 100. An individual score can be one of many values from 0 to 100.
2. The 6 outcomes from rolling one die. An individual outcome can be any 1 to 6 values.
3. The 4 martial status: single, married, widowed, divorced. An individual can be 1 of 4 status.
Or some variables have only 2 values or outcomes such as
1. Answer in a true false test.
2. Answer to yes/No questions in a questionnaire.
3. Manufactured product as defective or non defective.
4. Outcomes of tossing a coin (head / tail).
5. Outcomes of a dice game in terms of rolling a 7 or not rolling a 7.
6. Sex of new born infant (male/female)
If a variable has only 2 possible outcomes and if the probabilities of these outcomes do not change for each trial regardless of what has happened on previous trials then the variable is called a binomial variable. We could not consider the variable whose outcomes are rain and no rain to be a binomial variable, since the probabilities of rain changes every day, however , the result of tossing a coin is a binomial variable since the probabilities for heads and tails remain the same for each toss.
Now to utilize the binomial distribution, it is necessary to satisfy certain assumptions. These assumptions are as follows.
1. The experiment consist of n repeated trials.
2. Each trial has two possible outcomes, one called “success” and the other “failure”.
3.The probability of a success, denoted by p, is constant from one trial to the next.
3. 4. The repeated trials are independent of each other. To compute a binomial probability, it is necessary to specify, n, the number of trials, x, the number of success, the p, the probability of success on each trial.
Derivation: Let an experiment be repeated n times and let the occurrence of the event be called a success (S) and non-occurrence a failure (F), that is the result of each trial is classified as a success or a failure. A trial having only tow possible results, success and failure, is called a Bernoulli Trial. Then Bernoulli Trials, give rise to different number of successes and failures. Let p be the probability of a success and q be the probability of a failure in a single trial, so that p + q =1.
Furtehrmore, we assume that (i) for each Bernoulli trial, p, the probability of success, remains the same, and (ii) the successive trials are independent.
If X denotes the number of successes, then obviously X can take any one of the (n+1) values 0,1,2,...., n. Such an experiement is called a Binomial experiment and X is known as a binomial random variable. Now we desire to find the probabilities for each of these values of X. Since the trials are independent, the multiplication law of probability holds. Thus,