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Engineering Statistics ECIV 2305

Engineering Statistics ECIV 2305. Chapter 3 DISCRETE PROBABILITY DISTRIBUTIONS 3.1 The Binomial Distribution 3.2 The Geometric Distribution 3.3 The Hypergeometric Distribution 3.4 The Poisson Distribution. Engineering Statistics ECIV 2305. Section 3.2 The Geometric Distribution.

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Engineering Statistics ECIV 2305

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  1. Engineering Statistics ECIV 2305 Chapter 3 DISCRETE PROBABILITY DISTRIBUTIONS • 3.1 The Binomial Distribution • 3.2 The Geometric Distribution • 3.3 The Hypergeometric Distribution • 3.4 The Poisson Distribution 3.2 The Geometric Distribution

  2. Engineering Statistics ECIV 2305 Section 3.2 The Geometric Distribution 3.2 The Geometric Distribution

  3. Recall that the Binomial distribution is the distribution of the number of successes occurring in a fixed number of trials n. • Suppose that our interest is to count the number of Bernoulli trials performed until the first success occurs. Such random variable is said to have a geometric distribution with parameter p. • The pmf of the geometric distribution is: P(X = x) = (1 – p)x-1p E(X) = 1/p , Var(X) = (1 – p)/p2 • The cdf of the geometric distribution is: P(X≤x) = 1 – (1 – p)x 3.2 The Geometric Distribution

  4. Examples • What is the probability that a Head is obtained for the first time on the fourth coin toss? • P(X = 4) = (1 – p)4-1p = (1 – 0.5)3(0.5) = 1/16 • What is the probability that a Head is obtained for the first time on the first coin toss? • What is the probability that a Head is obtained for the first time on the first coin toss? 3.2 The Geometric Distribution

  5. Example • Draw the pmf of a geometric distribution with parameter p=1/2 and the pmf of a geometric distribution with parameter p=1/5 . (Fig. 3.12 and 3.13 in your textbook) 3.2 The Geometric Distribution

  6. Example Air Force Planes (page 175) • What is the distribution of the number of attempts needed to start a plane’s engine? • This is a case of a geometric distribution since attention is directed at the number of trials until the first success. A “success” here is the event that the plane’s engines start ►► p = 0.75 3.2 The Geometric Distribution

  7. Example Telephone Ticket Sales (page 176) • People make telephone calls to a salesperson to buy tickets for an event. The probability that a salesperson becomes free from the previous customer is 0.1. In other words, P(reaching a salesperson) = 0.1 • Find the distribution and the expectation of the number of calls that a person needs to make until a salesperson is reached? • What is the probability that a caller gets through on the fifth attempt? • What is the probability that 15 or more calls are needed? • The placing of a call is a Bernoulli trial with a success probability of p=0.1 • The geometric distribution is appropriate since the quantity of interest is the number of calls made until the first success. 3.2 The Geometric Distribution

  8. … continuedTelephone Ticket Sales … 3.2 The Geometric Distribution

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