Title Discrete Distributions.pdf ✅

Negative Binomial Distribution Consider an experiment with similar properties as those of binomial experiment, with an exception that trial are to be continued till desired number of success occurs. We are interested in the probability of kth success occurs in xth trial. Experiments of this kind are called negative binomial experiments. The number ܺ of trials to produce ݇ success in a negative binomial experiment is called negative binomial random variable and its probability distribution is called Negative Binomial Distribution Denoted by symbol ܤሺݔ ;݇, ݌ሻ . Now kth Success is to occur at xth trial, that means ሺ݇ െ 1ሻ success are to occur in ሺݔ െ 1ሻ trials. Hence, ܲሺܺൌݔሻ ൌ ൬ 1െݔ ݇െ1൰ ݌௞ݍ௫ି௞ , ݔ ൌ ݇ ൅ 1, ݇ ൅ 2, ... ෍ ݌ሺݔሻ ൌ෎ ൬ݔെ1 ݇െ1൰ ݌௞ݍ௫ି௞ ஶ ௫ୀ௞ ൌ ݌ ෎௞൬ݔെ1 ݔെ݇൰ ݍ௫ି௞ ஶ ௫ୀ௞ ݊ ൌ ݇ െ ݔ ݐ݈݁ ൌ ݌ ෍ ௞൫௡ା௞ିଵ ௡ ൯ݍ ௡ஶ ௡ୀ଴ ൌ ݌൛௞1 ൅ ൫௞ ଵ൯ݍ ൅ ൫௞ାଵ ଶ ൯ݍଶ ൅⋯ൟ ൌ ݌௞ሺ1 െ ݍሻି௞ ൌ 1 Moment generating function of negative Binomial Distribution ܯ௑ሺݐሻ ൌ ∑ ݌ሺݔሻ݁௫௧ ൌ ݌ ෍ ௞൫௡ା௞ିଵ ௡ ൯ݍ ௡ஶ ௡ୀ଴ ݁௡௧ ൌ ݌ ෍ ௞൫௡ା௞ିଵ ௡ ൯ሺݍ݁௧ሻ௡ ஶ ௡ୀ଴ ൌ ݌௞ሺ1 െ ݍ݁௧ሻି௞ Expectation: ܧሺܺሻ ൌߤൌ ௗ ௗ௧ ܯ௑ሺݐሻ|௧ୀ଴ ൌ ݌௞݇ݍ݁௧ሺ1 െ ݍ݁௧ሻି௞ିଵ|௧ୀ଴ ൌ ݍ݇௞݌ ሺ1െݍሻ௞ାଵ ൌ ݇ݍ ݌ Variance: ܧሺܺଶሻ ൌ ௗమ ௗ௧మ ܯ௑ሺݐሻ|௧ୀ଴ ൌ ݇݌ݍ௞ሾ݁௧ሺ1 െ ݍ݁௧ሻି௞ିଵ ൅ ݍ݁ଶ௧ሺ݇൅1ሻሺ1 െ ݍ݁௧ሻି௞ିଶሿ௧ୀ଴ 1 ቆ ݍ௞݌݇ ൌ ݌௞ାଵ ൅ ݍሺ݇൅1ሻ ݌௞ାଶ ቇ ൌ ݇ݍ ݌ ൅ ݇ሺ݇൅1ሻݍଶ ଶ݌ Hence, ߪଶሺܺሻ ൌ ܧሺܺଶሻ െ ሼܧሺܺሻሽଶ ൌ ௞௤ ௣మ