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Understanding Discrete Probability Distributions for Random Variables

Learn about Bernoulli distribution, PMF, CDF, and their properties with practical examples. Explore scenarios like engine tune-ups and telephone line usage to grasp the concepts effectively.

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Understanding Discrete Probability Distributions for Random Variables

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  1. Statistics 270 - Lecture 8

  2. Last day/Today: Discrete probability distributions • Assignment 3: Chapter 2: 44, 50, 60, 68, 74, 86, 110

  3. Example • Bernoulli Distribution: • X takes on two possible values: • p(x)=px(1-p)1-x • This is the probability distribution function or probability mass function • p is called a: • The collection of all pdf’s for different values of p, for example, is called

  4. Example (Chapter 2 – 11) • A garage specializing in engine tune-ups knows that 45% of all tune-ups are done on 4 cylinder vehicles…40% on 6 cylinder cars and the rest are eight cylinder cars • What is the pdf (pmf)? • What is the probability that a randomily selected car has at least 6 cylinders • What is the probability that the car has at most 6 cylenders

  5. Cumulative Distribution Function (cdf): The cdf of a discrete rv with pmf p(x) is defined, for each x, by

  6. Properties of the cdf:

  7. Example • Have 3 flips of a coin • X=number of heads observed • p(x)= • F(x)=

  8. Example • Plot of cdf

  9. Example (Chapter 2 – 13) • A mail order company has 6 telephone lines • Let X denote the number of lines in use at a specific time • The pmf for X is: • What is the probability that between 2 and 5 lines (inclusive) are active?

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