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A second-stage smog alert has been called in a certain area of Los Angeles County in which there are 50 industrial firms. An inspector will visit 10 randomly selected firms to check for violations of regulations. a. If 15 of the firms are actually violating at least one regulation, what is the pmf of the number of firms visited by the inspector that are in violation of at least one regulation? b. If there are 500 firms in the area, of which 150 are in violation, approximate the pmf of part (a) by a simpler pmf. c. For X = the number among the 10 visited that are in violation, compute E(X) and V(X) both for the exact pmf and the approximationg pmf in part (b).Let X be the number of material anomalies occurring in a particular region of an aircraft gas- turbine disk. The article "Methodology for Probabilistic Life Prediction of Multiple- Anomaly Materials" (Amer. Inst. of Aeronautics and Astronautics J., 2006: 787-793) proposes a Poisson distribution for X. Suppose that p = 4. a. compute both P (X = 4) and P ( X 0 a. This pdf is a member of what family introduced in this chapter? b. If o = 20mm (close to the value suggested in the paper), what is the probability that a dart will land within 25 mm (roughly 1 in.) of the target?Let X denote the lifetime of a component, with f(x) and F(x) the pdf and cdf of X. The probability that the component fails in the interval (x, x + Ax) . Dividing this by produces the failure rate function: Ax An increasing failure rate function indicates that older components are increasingly likely to wear out, whereas a decreasing failure rate is evidence of increasing reliability with age. In practice, a "bathtub-shaped" failure is often assumed. a. If X is exponentially distributed, what is /(x)? b. If X has a Weibull distribution with parameters a and B,what is /(x)? For what parameter values will (x) be increasing? For what parameter values will r(x) decrease with x? f(x) r(x) = 1 - F(x) so that if a component lasts hours, it will last B forever (while seemingly unreasonable, this model can be used to study just "initial wearout"). What are the cdf and pdf of X