Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Statistics 400
Document related concepts
Transcript
Statistics 270 - Lecture 10 • Last day/Today: Discrete probability distributions • Assignment 3: Chapter 2: 44, 50, 60, 68, 74, 86, 110 Common Discrete Probability Distributions • There are several probability distributions that describe a large variety of random phenomena • Will consider 5 of these: 1. 2. 3. 4. 5. Discrete Uniform Bernoulli Binomial Hypergeometric Poisson Discrete Uniform Distribution • Have seen this already • Random variable X has k possible outcomes, each equally likely • pmf: • Mean and Variance: Bernoulli Distribution • Have seen this already • Random variable X has 2 possible outcomes • pmf: • Mean and Variance: Binomial Distribution • Count the number of successes in n independent Bernoulli trials • Binomial Experiment: • • • • Have n trials, where n is fixed in advance of the experiment Each trial results in one of two possible outcomes (success or failure) The outcomes are independent The probability of success is constant Binomial Distribution • Count the number of successes in n independent Bernoulli trials • Binomial Experiment: • • • • Have n trials, where n is fixed in advance of the experiment Each trial results in one of two possible outcomes (success or failure) The outcomes are independent The probability of success is constant Binomial Distribution • Let X denote the number of successes in n independent Bernoulli trials • Then the rv, X, is said to be a binomial rv • pmf Binomial Distribution • Mean: • Variance: Example • A baseball player has a 300 batting average • What is the expected number of hits in 25 at bats? Example • According to CTV News, the 2006 Federal Election results were: Party Conservative Liberal NDP Bloc Quebecois Independent Green Other % of Popular Vote 36.3 30.2 17.5 10.5 0.6 4.5 0.4 Example • Ten voters from across the country are randomly selected and the number of of Conservative voters is counted • • Is this a Binomial experiment? • What is the probability that 6 of them voted for the Conservatives? • What is the expected number of Conservative votes in such a sample? Hyper-geometric Distribution • Count the number of successes in n trials from a population with N individuals and M successes • Assumptions: • • • • Have n trials, where n is fixed in advance of the experiment Population has N individuals (finite population) There are two possible outcomes (success and failure) and there are M successes in the population A sample of n individuals is taken WITHOUT replacement Hyper-geometric Distribution • Let X denote the number of successes in a sample of size n, without replacement • Then the rv, X, is said to be a hyper-geometric rv • pmf Hyper-geometric Distribution • Mean: • Variance: Example (Chapter 3 - 64) • A digital camera comes in either a 3 or 4 mega-pixel version • A store receives 15 cameras and 6 are the 3 mega-pixel version • Suppose 5 of these are randomly selected and stored behind the counter • Let X denote the number of 3 mega-pixel cameras stored behind the counter • Compute P(X=2) and P(X <=2)
