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Chapter 3b

# Chapter 3b - USING THE BINOMIAL CUMULATIVE DISTRIBUTION...

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USING THE BINOMIAL CUMULATIVE DISTRIBUTION FUNCTION TO FIND PROBABILITIES The cumulative distribution function represents P(X < x). The binomial tables present the CDF of the binomial distribution for various values of n, p, and x. Let n =4 and p = .2 Find P(X < 3). Find P(X = 2) Find P(X > 2)

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NEGATIVE BINOMIAL PROBABILITY DISTRIBUTION Conditions: The number (n) of trials is fixed. The number of successes will vary from experiment to experiment. The probability of getting a success, p, is constant throughout the trials. The trials are observed until exactly r successes are obtained (r is a positive integer set by the experimenter). The random variable X is the number of trials needed to obtain the r successes. Negative Binomial Probability Function: For r = 1, 2, 3, … and x = r, r+1, r+2, … 1 ( ) (1 ) 1 r x r x P x p p r - -   = -   -   Binomial vs. Negative Binomial Distribution: Binomial: What is the probability of 2 successes out of 3 trials? Number of trials is fixed but numbers of successes can vary. Negative Binomial: What is the probability that it will take three trials to get two successes? Number of successes is fixed but number of trials to get those successes can vary.
Mean and Variance of Negative Binomial distribution E(X) = r/p Var(X) = r(1-p)/p 2 Shape of the distribution depends on r and p: All plots shown here with mean of 10

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Chapter 3b - USING THE BINOMIAL CUMULATIVE DISTRIBUTION...

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