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| Binomial Distribution |
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| A trial, which has only two outcomes i.e., "a success" or "a failure", is called a Bernoulli trial. |
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| Let X be the number of successes in a Bernoulli trial, then X can take 0 or 1 and |
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| P(X =1) = p = "probability of a success" |
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| P(X = 0) = 1 - p = q = "probability of failure". |
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| Consider a random experiment of performing n independent Bernoulli trials. |
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| Let p be the probability of success, q = 1 - p be the probability of failure. |
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| The probability of x successes and consequently (n - x) failures in n independent trials in a specified order say SSSFFSSFF….FSF is given by |
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| P(SSSFFSSFF…FSF) |
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| = P(S) P(S) P(S) P(F) P(F) P(S) P(S) P(F)….P(F) P(S) P(F) |
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| = p.p.p.qq.ppq….qpq |
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| = pxqn-x |
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| But x successes can occur in nCx ways. |
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| \ P(X = x) = nCx px qn-x is the probability mass function of exactly x successes. |
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| The probability distribution of the number of successes, so obtained is called the binomial distribution. |
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| X P[X = x] |
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| 0 qn |
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| 1 nC1 pqn-1 |
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| 2 nC2 p2 qn-2 |
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| 3 nC3 p3 qn-3 |
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| . . |
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| . . |
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| . . |
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| . . |
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| n pn |
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| Note 1: |
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| n and p are called the parameters of the binomial distribution. |
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| Note 2: |
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| If x is a binomial variate with parameters n and p then it is denoted by x = b(n, p). |
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| Note 3: |
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| Example: |
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| 5 cards are drawn successively with replacement from well shuffled deck of 52 cards. What is the probability that |
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| i) all the five cards are spades |
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| ii) only 3 cards are spades |
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| iii) none is a spade. |
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| Suggested answer: |
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| Let X be the random variable for the number of spade cards drawn. |
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| p = probability of drawing a spade card |
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| q = 1 - p |
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| n = 5 |
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| We have |
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| P(X = x + 1) = nCx+1 px+1 qn-x-1 |
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| = np (p+q)n-1 |
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| To find the variance: |
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| We have |
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| = n(n-1) p2 (p+q)n-2 + np |
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= n2p2 - np2 + np  |
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| = n2p2 + np(1-p) |
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| = n2p2+ npq |
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| Now V(x) = E(x2) - [E(x)]2 |
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| = n2p2 + npq - n2 p2 |
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| = npq |
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| Example: |
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| If the mean and variance of a binomial distribution are respectively 9 and 6, find the distribution. |
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| Suggested answer: |
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| Mean of a binomial distribution = np = 9 |
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| Variance of a binomial distribution = npq = 6 |
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| Binomial distribution is given by |
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