Some Spcial Distributions

The Negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs. For example, if we define a "1" as failure, and all non "1"s as successes, and we throw a die repeatedly until the third time “1” appears (r = three failures), then the probability distribution of the number of non-“1”s that had appeared will be negative binomial.
Suppose there is a sequence of independent Bernoulli trials, each trial having two potential outcomes called “success” and “failure”. In each trial the probability of success is p and of failure is (1 ? p). We are observing this sequence until a predefined number r of failures has occurred. Then the random number of successes we have seen, X, will have the negative binomial (or Pascal) distribution:

When applied to real-world problems, outcomes of success and failure may or may not be outcomes we ordinarily view as good and bad, respectively. Suppose we used the negative binomial distribution to model the number of days a certain machine works before it breaks down. In this case “success” would be the result on a day when the machine worked properly, whereas a breakdown would be a “failure”. If we used the negative binomial distribution to model the number of goal attempts a sportsman makes before scoring a goal, though, then each unsuccessful attempt would be a “success”, and scoring a goal would be “failure”. If we are tossing a coin, then the negative binomial distribution can give the number of heads (“success”) we are likely to encounter before we encounter a certain number of tails (“failure”). In the probability mass function below, p is the probability of success, and (1-p) is the probability of failure.