By Guderley K. G., Keller C. L.

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3 Examples of the binomial distribution for different success probabilities. 90 of curing a disease. It is administered to 100 patients, where the outcome for each patient is either cured or not cured. 90). • The National Institute of Mental Health estimates that there is a 20% chance that an adult American suffers from a psychiatric disorder. Fifty adult Americans are randomly selected. 20). • A manufacturer of computer chips finds that on the average 5% are defective. To monitor the manufacturing process, they take a random sample of size 75.

The Poisson distribution is often used to approximate the binomial. When n is large and p is small (so np is moderate), then the number of successes occurring can be approximated by the Poisson random variable with parameter λ = np . The Poisson distribution is also appropriate for some applications where events occur at points in time or space. We see it used in this context in Chapter 12, where we look at modeling spatial point patterns. Some other examples include the arrival of jobs at a business, the arrival of aircraft on a runway, and the breakdown of machines at a manufacturing plant.

A. What is the mean lifetime of the flat panel display? b. What is the probability that the display fails within the first two years? c. Given that the display has been operating for one year, what is the probability that it will fail within the next year? 7. The time to failure for a widget follows a Weibull distribution, with ν = 0 , β = 1 ⁄ 2 , and α = 750 hours. a. What is the mean time to failure of the widget? b. What percentage of the widgets will fail by 2500 hours of operation? That is, what is the probability that a widget will fail within 2500 hours?

### A Basic Theorem in the Computation of Ellipsoidal Error Bounds by Guderley K. G., Keller C. L.

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