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Probability Distribution Binomial Probability Distribution Dr. Vlad Monjushko, PhD, MsC

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Discrete Random Variable (DRV) Probability Distribution Discrete probability distribution includes all the values of DRV; Discrete probability distribution includes all the values of DRV; For any value x of DRV: 0P(x)1; For any value x of DRV: 0P(x)1; The sum of probabilities of all the DRV values equals to 1; The sum of probabilities of all the DRV values equals to 1; The values of DRV are mutually exclusive. The values of DRV are mutually exclusive. Properties:

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Binomial Random Variable Binomial Random Variable (BRV) probability distribution is a special case of the DRV probability distribution, when there are only two outcomes: Success and Failure; Binomial Random Variable (BRV) probability distribution is a special case of the DRV probability distribution, when there are only two outcomes: Success and Failure; BRV value is defined as the number of Successes in a given number of trials; BRV value is defined as the number of Successes in a given number of trials; Conditions that must be met to consider the experiment as Binomial: –Every trial must have only two mutually exclusive outcomes: Success or Failure, –The probability of Success and Failure must remain constant from trial to trial, –The outcome of the trial is independent of the outcomes of the previous trials.

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Probability of x Successes in n trials P – probability of x-successes in n-trials; x – number of successes; n – number of trials; p – probability of a success in one trial; The results of the calculation for the values of n 20 could be also found in Binomial Probabilities table; For the values of n>20 we use Normal Approximation of Binomial Distribution.

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x P Example For the number of trials n=5, and for a probability of one success p=0.5, the binomial probability distribution has the following shape:

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Statistical Parameters of a Binomial Distribution Mean of the distribution: Mean of the distribution: µ=n. p Variance: Variance: σ 2 =n. p. (1-p) Standard Deviation: Standard Deviation: σ=n. p. (1-p) n – number of trials; p – probability of a success in one trial;

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Summary Binomial Random Variable (BRV) is a specific case of a Discrete Random Variable, when there are only two outcomes: Success and Failure. Binomial Random Variable (BRV) is a specific case of a Discrete Random Variable, when there are only two outcomes: Success and Failure. BRV value is defined as the number of Successes in a given number of trials. BRV value is defined as the number of Successes in a given number of trials. Probability of any value of a BRV for a small sample case ( could be either calculated or found in the Probability of any value of a BRV for a small sample case (n 20) could be either calculated or found in the Binomial Probabilities table. For a large sample case (n>20) we use Normal Approximation of a Binomial Distribution.

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