State the binomial theorem and explain its role in probability and statistics

Running Head : BINOMIAL THEOREM IN STATISTICS

State the binomial theorem and explain its role in probability and statistics

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State the binomial theorem and explain its role in probability and statistics

The Binomial Theorem states that for any positive integer n , the expansion of the binomial expression (x a )n is given by

The Binomial Theorem , although has been developed deeply in the field of Combinatorics , we cannot deny the indispensable use of this in Statistics and Probability Theory (1

1 )n 2n by the Binomial Theorem . In combinatorics , there are still many identities that can be proven through the use of the binomial theorem , and those identities , like the one above has a very vital role in probability

Another good application of the theorem is in finding the probability of an event when the distribution has a binomial distribution . A distribution is said to be binomial if for a finite number of trials each trial has only two outcomes . As an illustration , tossing a coin m times is an experiment with binomial distribution . In probability theory , if X is the random variable and is defined as the number of heads that appears after m tosses , and

is the probability of having a head in a toss , then

Note that the formula above is just the same as the Binomial Theorem stated above with a

and x (1 -

. But this time , we must also recognized that the index of the summation is not the same as upper index of the coefficient of pr (1-p )m-r . This simply means that we are only solving for the first (n 1 ) terms of the binomial expansion and not up to (m 1 ) terms or not the whole expansion . In other words , the formula above can be interpreted as the probability of having n heads after two tosses is equal to the binomial expansion after (n 1 ) term . To further explain this , consider the next problem

Problem . Ten students are going to take an exam . If the probability that a student will pass the exam is 0 .8 , then what is probability that at most half of them will fail the exam

Solution . Let X number of students who will fail the exam

p probability that a student will fail the exam 0 .2 (since to pass 0 .8

We want

(X 5

Since all the needed information are given , then all we need to do is apply the theorem (1 (0 .107374 (10 (0 .026843 (45 (0 .006711 (120 (0 .001678 (210 (0 .000419 (252 (0 .000105 0 .993631

Thus , the probability that at most 5 students will fail the exam is 0 .993631

Aside from the mathematical value of summation that is linked to statistics and probability , and its application in problems on binomial distribution , the theorem is also a powerful tool to derive formulas for moments of other distributions that uses the basic framework of the binomial theorem . In other...