1. (a) Y1,Y2, ... , Yn form a random sample from a probability distribution with cumu-lative distribution function Fy (y) and probability density function fy (y). LetY(1) = min{Y1, Y2,..., Yn}.Write the cumulative distribution function for Y(1) in terms of Fy(y) and henceshow that the probability density function for Y(1) isfY, (y) = n{1 – Fy(y)}"-' fy(y).(b) An engineering system consists of 5 components connected in series, so, if onecomponents fails, the system fails. The lifetimes (measured in years) of the 5components, Yı, Y2, . . . , Y5, are all independent and identically distributed.(i) Suppose the lifetimes follow the standard uniform distribution U (0, 1). Findthe probability density function for Y(1), the time to failure for the system,and hence find the probability that the system functions for at least 6months without failing.(ii) If, instead, the lifetimes follow an exponential distribution with mean 0,then Y(1) follows an exponential distribution with mean 0/5. Prove thisresult.Assuming that the only information available is a single observation on Y(1),find the most powerful test of size 0.05 for Ho : 0 = 01 versus H1 : 0where 01 < 02. (Hint: the probability density function and cumulativedistribution function for an exponential random variable with mean 0 aref (y) = 0-1 exp(-y/0), y > 0, and F(y) = 1 – exp(-y/0), y > 0, respec-tively.)02,||

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(a) Y1, Y2, ..., Yn form a random sample from a probability distribution with cumu- lative distribution function Fy(y) and probability density function fy(y). Let Y(1) = min{Y1, Y2,..., Yn}. Write the cumulative distribution function for Y(1) in terms of Fy(y) and hence show that the probability density function for Y(1) is fy, (y) = n{1– Fy(y)}"-'fy(y).

(a) Y1, Y2, ..., Yn form a random sample from a probability distribution with cumu- lative distribution function Fy(y) and probability density function fy(y). Let Y(1) = min{Y1, Y2,..., Yn}. Write the cumulative distribution function for Y(1) in terms of Fy(y) and hence show that the probability density function for Y(1) is fy, (y) = n{1– Fy(y)}"-'fy(y).

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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