jbstatistics · YouTube · 7:32
Contrast fixed-trial binomial counts with the waiting time to a fixed number of successes in the negative binomial model.
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A binomial model fixes the number of trials and counts successes. Here the negative binomial variable instead counts the trials needed to reach r successes, with independent trials and a constant positive success probability p.
To finish on trial x, the first trials contain exactly successes, and trial x succeeds. Thus the probability is ʳ()^(), for integers .
Under the total-trials convention used here, the mean is . With and , it is 33⅓ trials. A failures-only convention shifts the variable and its mean by r.
A binomial model fixes the number of trials and counts successes. Here the negative binomial variable instead counts the trials needed to reach r successes, with independent trials and a constant positive success probability p.
To finish on trial x, the first trials contain exactly successes, and trial x succeeds. Thus the probability is ʳ()^(), for integers .
Under the total-trials convention used here, the mean is . With and , it is 33⅓ trials. A failures-only convention shifts the variable and its mean by r.
A binomial model fixes the number of trials and counts successes. Here the negative binomial variable instead counts the trials needed to reach r successes, with independent trials and a constant positive success probability p. To finish on trial x, the first trials contain exactly successes, and trial x succeeds. Thus the probability is ʳ()^(), for integers . Under the total-trials convention used here, the mean is . With and , it is 33⅓ trials. A failures-only convention shifts the variable and its mean by r.
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