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Probability Distributions

Probability Distributions. A random variable is a variable (letter) whose values are determined by chance It can be continuous (something that’s measured) or discrete (something that’s counted) Continuous examples: temperature, weight Discrete examples: numbers on dice

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Probability Distributions

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  1. Probability Distributions • A random variable is a variable (letter) whose values are determined by chance • It can be continuous (something that’s measured) or discrete (something that’s counted) • Continuous examples: temperature, weight • Discrete examples: numbers on dice • A discrete probability distribution consists of: • The values a random variable can take • The corresponding probabilities

  2. Probability Distribution Examples • Rolling one die and looking at the number • Outcomes are 1, 2, 3, 4, 5, 6 • Probabilities are 1/6, 1/6, 1/6, 1/6, 1/6, 1/6 • Tossing 3 coins and counting number of heads • Outcomes are 0, 1, 2, 3 • Probabilities are 1/8, 3/8, 3/8, 1/8 • Selecting one card from a deck, looking for a spade • Outcomes are “spade,” “not a spade” • Probabilities are 13/52 and 39/52

  3. Probability Distribution Requirements • The sum of the probabilities of all the events in the sample space must be 1: ∑P(X)=1 • The probability of each event in the sample space must be between 0 and 1, inclusive: 0 ≤ P(X) ≤ 1 for all X

  4. Statistics of a Probability Distribution • Mean: µ=∑X•P(X) • Expected value: Another word for the mean, written E(X) • Example: Find the expected value of the gain (amount of money you make) if you buy a lottery ticket

  5. Binomial Distribution • A binomial experiment is a special kind of probability experiment, with the following requirements: • A fixed number of trials • Each trial has two outcomes, designated success and failure • The outcomes of the trials are independent • The probability of success is the same for each trial • The outcomes and probabilities associated with a binomial experiment are called a binomial distribution

  6. Binomial Distribution • p = Probability of success in one trial • q = Probability of failure in one trial • Note that q = 1 - p • n = Number of trials • X = Number of successes in n trials • Note that 0 ≤ X ≤ n • In a binomial experiment, the probability of exactly X successes in n trials is:

  7. Binomial Distribution • You do not need to use the formula to calculate the binomial probabilities • You can read the probabilities from a table • Use Table B in Appendix C, starting on page 626 • Specify n, X, and p • Add or subtract table values to get “at most” or “at least” answers

  8. Statistics for the Binomial Distribution • Mean = µ • Standard deviation = 

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