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38 Binomial Experiments-I 1 Binomial experiments are those for which the outcome from each trial is one of only two options (“success” or “failure”). The properties of a binomial experiment are: 1. The experiment is repeated for a fixed number of trials, where each trial is independent of all the others. 2. There are only two possible outcomes for each trial. The outcomes can be classified as a success (S) or a failure (F). 3. The probability of success is the same for all trials. 4. The random variable X counts the number of successful trials out of n trials. 38 Binomial Experiments-II 1 Why is the following a binomial experiment? We randomly sample 500 fish from the population. We record whether each animal is mature or immature. The random variable X is the number of mature animals. Binomial Experiments-II 38 (Notation) 1 1. The experiment is repeated for a fixed number of trials 2. There are only two possible outcomes (S and F) 3. The probability of success P(S) is the same for each trial 4. The random variable x counts the number of successful trials n The number of times a trial is repeated New p P(S) The probability of a success in a single trial terms qP(F) The probability of a failure in a single trial,q 1- p x The number of successes on n trials (x=0,1,..,n) 38 Binomial Probabilities-I 1 In a binomial experiment, the probability of exactly x successes in n trials is: P(x) C pxqnx n! pxqnx n x (n x)!x! The binomial probability therefore involves the probability of x successes and n -x failures multiplied by the number of ways choosing x successes out of n trials. n and p are known as parameters. Much of statistics involves using data to estimate the values for unknown parameters. 38 Binomial Probabilities-II 1 Notation: X ~ B(n; p) We read this as “The random variable X is distributed binomially with parameters n and p”. Mean: np Variance:2 npq
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