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# week5class - Probability Models A Bernoulli Trial A random...

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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

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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?

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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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