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Unformatted text preview: Probability Models A Bernoulli Trial A random variable X is a Bernoulli random variable (or Bernoulli trial) if the following conditions are met: X has only two possible outcomes (called success and failure) The probability of success, p , is constant for each trial The trials are independent. Examples of Bernoulli random variables: The Geometric Model Suppose that repeated Bernoulli Trials each having probability of success p are performed until a success occurs. If we denote with X the number of trials required until a first success occurs then X is a Geometric random variable and The expected number of trials required until a success is observed (expected value) for a Geometric random variable is Example: Lets go back in time to the spring of 2000. How many girls is Adam going to have to ask until 1 of them agrees to go to the senior prom with him? The first answer that comes to mind is 0 since Adam is so cool that all the girls would be asking him to the prom. Lets assume though, that Adams true success rate with ladies is 20% and that this percent is constant for all girls Adam asks. Whats the probability that The fifth girl Adam asks is the first girl to say yes? (or say fine, whatever.) Adam gets a yes in three attempts or less? Whats the expected number of girls Adam will have to ask until one will go to the prom with him? If Adam can take multiple girls to the prom, whats the probability that the first girl to say no to him is the fourth girl he asks? Binomial Distributions Binomial distributions are models for some categorical variables, typically representing the number of successes in a series of n trials. Observations must meet the following requirements in order for the overall process to be deemed binomial: The total number of observations, n, must be fixed in advance. Each observation has only two possible outcomes, success ( S ) and failure ( F ). All n observations must have the same probability of success, p . The observations are all independent of each other, meaning the outcome of each observation is not affected by the outcomes of other observations. Example: Randomly draw n balls with replacement from an urn containing 10 red balls and 20 black balls. Use S to denote the outcome of drawing a red ball and F to denote the outcome of a yellow ball (here we are defining a red ball to be a success and a yellow ball to be a failure). Question : Would it still be a binomial experiment if the balls were drawn without replacement? Binomial Random Variables & Distributions Binomial Random Variable A binomial random variable counts the number of successes in n trials of the binomial experiment....
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This note was uploaded on 11/28/2009 for the course STATS STATS10 taught by Professor Sugano during the Spring '09 term at UCLA.
 Spring '09
 SUGANO
 Bernoulli, Probability

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