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**Probability Distribution**

Binomial Probability Distribution Dr. Vlad Monjushko, PhD, MsC

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**Discrete Random Variable (DRV) Probability Distribution**

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

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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; 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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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: P x

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**Statistical Parameters of a Binomial Distribution**

Mean of the distribution: µ=n.p Variance: σ2=n.p.(1-p) 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. 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 (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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