18 chapter 5

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Unformatted text preview: ion function of X is defined as: P(x ) = P( x successes in n independent trials) for x = 0, 1, 2, . . . , n This is known as the binomial distribution. 18 Chapter 5 A calculation formula for the probabilities can be obtained as follows. In n independent trials, the probability of x successes and (n−x) failures is: p x (1 − p)n− x The number of combinations of x successes in n trials is: Cn = x n! x ! (n − x) ! Therefore, the probability distribution function for the binomial distribution is: 19 P(x ) = n! p x (1 − p)n − x x ! (n − x) ! for x = 0, 1, 2, . . . , n Chapter 5 Note: a calculation rule is 0! = 1 Examples of application of the binomial distribution are: P(X = 0) = P( no successes in n trials ) = (1 − p)n P(X = 1) = P( one success in n trials ) = n p (1 − p)n − 1 The cumulative probability function of the binomial distribution may have useful application and is calculated as: F( x ) = P( X ≤ x ) for x = 0, 1, 2, . . . , n For example, F(1) = P(X ≤ 1) = P( at...
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