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Lec06 - Ch 3 Learning Objectives IOE/Stat 265 Fall 2009...

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1 IOE/Stat 265, Fall 2009 IOE/Stat 265, Fall 2009 Lecture #6: Lecture #6: Discrete Distributions Discrete Distributions 3-1 Discrete Random Variables 3-2 Probability Distributions for Discrete Random Variables 3.3 Expected Values 3.4 Binomial Probability Distribution 3.5 Hypergeometric and Neg. Binomial 3.6 Poisson Probability Distribution 2 Ch 3: Learning Objectives 1. Determine probabilities from PMF and vice-versa 2. Determine probabilities from CDF and determine CDF from PMF 3. Calculate means and variances for discrete r.v. 4. Learn some basic discrete PMFs and their applications 5. Calculate probabilities, means, and variances for select discrete distributions. 3 Recall: Random Variables Definition: A random variable (r.v.) is a function (rule) that assigns a real number to each possible outcome in the sample space of a random experiment. Example: if X is a random variable that defines the number of heads that we may obtain when we flip 3 coins, then X can take on 4 possible values: 0, 1, 2, 3 A r.v. is denoted “X”. The measured random variable (aka realization) is denoted “x” 4 3-1 Discrete or Continuous Variables A discrete r.v. has finite range e.g. number of scratches in paint, proportion of defects in 10 samples, number of errors. A continuous r.v. is defined on an interval of real numbers. e.g. pressure, temperature, voltage, current, weight. This chapter confines attention to Discrete case versus
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5 Example: Sum of Two Dice Let X be a random variable that represents the sum of rolling two dice. Fill in the blanks below with the values of X and its frequency… Combination X Frequency (1,1) (1,2) (2,1) (1,3) (3,1) (2,2) (1,4) (4,1) (2,3) (3,2) (1,5) (5,1) (2,4) (4,2) (3,3) (1,6) (6,1) (2,5) (5,2) (3,4) (4,3) (2,6) (6,2) (3,5) (5,3) (4,4) (3,6) (6,3) (4,5) (5,4) (4,6) (6,4) (5,5) (5,6) (6,5) (6,6) 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 6 Example 2: Bernoulli Process A special case of a discrete r.v. whose values are only 0 or 1 is called a Bernoulli random variable Common Notation: x = 0 Failure (F) x = 1
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