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Topic -Ve Binomial Distribution & Geometric Distribution. Class BS(CS) 2nd Semester Submitted To M.Yahya Submitted From M. Zishan Zafar, M. Waqas. Submission Date 17-08-2005 What is Distribution Statistical arrangement of values of a variable showing their observed or theoretical frequency of occurance. What is binomial distribution In probability, a binomial distribution gives the probabilities of k outcomes A (or nk outcomes B) in n independent trials for a two-outcome experiment in which the possible outcomes are denoted A and B. (OR) The theoretical frequency distribution of events that have two possible outcomes which are “success” or “failure” Explaination A probability distribution that applies to experiments involving sequences of independent trials in which only two possible outcomes (eg, success or failure) can result on each trial. If p is the probability of success on each trial, and q = 1 - p the probability of failure, then the probability of success occurring x times in n trials is given by the binomial distribution Negative Binomial Distribution Definition: If repeated independent trials can result in a success with probability p and a failure with probability q = 1-p, then the probability description of the random variable X the number of trials on which success occurs is given by b*(x; k, p) = pk qx-k, x = k, k+1, k+2, … Explanation This distribution is also called the Pascal distribution after the French mathematician Blaise Pascal (1623-1662) The distribution is found to occur in many biological situations and in inverse sampling from a binomial population. In binomial experiment, the number of successes varies and the number of trials is fixed. But there are the experiments in which the number of successes fixed and the number of trials varies to produce the fixed number of successes. Such an experiments are called negative binomial experiments. When denote the number of trials to produce k successes in a negative binomial experiment, it is called a negative binomial variable and its p.d. is called the negative binomial distribution. When the negative binomial r.v. X assumes a value x on which the kth success occurs, the negative binomial distribution is given by b*(x; k, p) = pk qx-k, x = k, k+1, k+2, … Conditions For Negative Binomial Distribution (1) The out comes of each trial may be classified into one or two categories: success (S) and failure (F). (2) The probability of success, denoted by P remains constant for all trials. (3) The successive trials are all independent. (4) The experiment is repeated a variable number of times to obtain a fixed number of successes. Example: Find the probability that a person tossing 3 coins will either all heads or all tails for the second time on the fifth toss. Solution: Now here x= 5, k = 2 and p = ¼ b*(x; k, p) = x-1 k-1 b*(5; 2, ¼) = 4 1 pk qx-k, 1 4 x = k, k+1, k+2, … 3 4 = 4!__ . 33 1! 3! 45 = 37 256 Geometric Distribution Definition If repeated independent trials can result in a success with probability p and a failure with probability q = 1-p then the probability distribution of the random variable X, the number of the trial on which the first success occurs, is given by g(x; p) = pqx-1 for x = 1, 2, 3, … when an experiment consists of independent trials with probability p of success and the trials are repeated until the first success occurs, it is called a geometric experiment. Conditions for Geometric Distribution (1) The outcomes of each trial may be classified into one of two categories, success and failure. (2) The probability of success p remains constant for all trials. (3) The success trials are all independent. (4) The experiment is repeated a variable number of times until the first success is obtained. If X represents the number of trials needed for the first success, then X is called a geometric r.v and its p.d. is called the geometric distribution. It has only one parameter p and is denoted by g(x ; p).The geometric distribution drive its name from the fact that its successive terms constitute a geometric progression. Since a Geometric r.v represents how long one has to wait for her success. Example: Find the probability that a person flipping a balanced coin requires 4 tosses to get a head. Solution: Here x= 4 and p = ½ using geometric distribution we have g(4 ; ½) = ½ (1/2)3 = 1/16 Summary : Now here we discuss two topics which are related to distribution. First we discuss about –ve binomial distribution a –ve binomial distribution occurs when (1) The out comes of each trial may be classified into one or two categories: success (S) and failure (F). (2) The probability of success, denoted by P remains constant for all trials. (3) The successive trials are all independent. (4) The experiment is repeated a variable number of times to obtain a fixed number of successes. And similarly a geometrical distribution occurs when (1) The outcomes of each trial may be classified into one of two categories, success and failure. (2) The probability of success p remains constant for all trials. (3) The success trials are all independent. (4) The experiment is repeated a variable number of times until the first success is obtained. Conclusion If repeated independent trials can result in a success with probability p and a failure with probability q = 1-p, then the probability description of the random variable X the number of trials on which success occurs is given by b*(x; k, p) = and pk qx-k, x = k, k+1, k+2, … If repeated independent trials can result in a success with probability p and a failure with probability q = 1-p then the probability distribution of the random variable X, the number of the trial on which the first success occurs, is given by g(x; p) = pqx-1 for x = 1, 2, 3, …