Bernoullis Random Variables And Binomial Distribution
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1.10 Bernoulli’s random Variables & Binomial Distribution<br />
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Bernoulli Random Variable<br />Suppose that a trial, or an experiment, whose outcome can be classified as either a success or a failure is performed. If we let X=1 when the outcome is a success and X=0 when the outcome is a failure, then the pmf of X is given by<br />
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Bernoulli Random Variable<br /> A random variable X is said to be a Bernoulli random variable (after the Swiss mathematician James Bernoulli) if its probability mass function is given by <br />
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Binomial Random Variable<br />Suppose now that n independent trials, each of which results in a success with probability p and in a failure with probability 1-p, are to be performed. If X represents the number of successes that occur in the n trials, then X is said to be a Binomial random variable with parameters (n,p) . Thus a Bernoulli random variable is just a binomial random variable with parameters (1,p) .<br />
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Binomial Distribution<br />Bernoulli Trials<br />There are only two possible outcomes for each trial.<br />The probability of a success is the same for each trial.<br />There are n trials, where n is a constant.<br />The n trials are independent.<br />
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Binomial Distribution <br />Let X be the random variable that equals the number of successes in n trials.<br />If p and 1 – p are the probabilities of success and failure on any one trial then the probability of getting x successes and n – x failures in some specific order is <br />px(1- p)n – x<br />The number of ways in which one can select the x trials on which there is to be a success is <br />
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Binomial Distribution <br />Thus the probability of getting x successes in n trials is given by<br />This probability distribution is called the binomial <br />distribution because for x = 0, 1, 2, …, and n the <br />value of the probabilities are successive terms of <br />binomial expansion of [p + (1 – p)]n;<br />
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Binomial Distribution <br />for the same reason, the combinatorial quantities <br />are referred to as binomial coefficients. <br />The preceding equation defines a family of probability distributions with each member characterized by a given value of the parameterp and the number of trials n.<br />
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Binomial Distribution <br />Distribution function for binomial distribution<br />
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Binomial Distribution <br />The value of b(x;n,p) can be obtained by formula<br />since the two cumulative probabilities B(x; n, p) and B(x - 1; n, p) differ by the single term b(x; n,p).<br />If n is large the calculation of binomial probability can become quite tedious.<br />
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Binomial Distribution Function<br />Table for n = 2 and 3 and p = .05 to .25<br />
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The Mean and the Variance of a Probability Distribution<br />Mean of discrete probability distribution<br />The mean of a probability distribution is the mathematical expectation of a corresponding random variable. <br />If a random variable X takes on the values x1, x2, …, or xk, with the probability f(x1), f(x2),…, and f(xk), its mathematical expectation or expected value is <br /> = x1· f(x1) + x2· f(x2) + … + xk· f(xk) <br />
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The Mean and the Variance of a Probability Distribution <br />Mean of binomial distribution<br />p probability of success<br />n number of trials<br />Variance of binomial distribution<br />
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The Mean and the Variance of a Probability Distribution <br />Mean of binomial distribution<br />p probability of success<br />n number of trials<br />Proof:<br />
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The Mean and the Variance of a Probability Distribution <br />Put x – 1= y and n – 1 = m, so n – x = m – y, <br />
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Computing formula for the variance<br />Variance of binomial distribution<br />Proof:<br />
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Put x – 1 = y and n – 1 = m <br />The Mean and the Variance of a Probability Distribution <br />
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The Mean and the Variance of a Probability Distribution <br />
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Put y – 1 = z and m – 1 = l in first summation<br />The Mean and the Variance of a Probability Distribution <br />
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Moment Generating function for Binomial distribution<br />
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Second ordinary/raw moment (moment about origin)<br />Moment Generating function for Binomial distribution<br />
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Moment Generating function for Binomial distribution<br />Moment Generating function for Binomial distribution<br />
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