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1. (a) Y1,Y2,...,Yn form a random sample from a probability distribution with cumulative distribution function FY...

1. (a) Y1,Y2,...,Yn form a random sample from a probability distribution with cumulative 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(1)(y) = n{1−FY (y)}n−1fY (y). [8 marks]

(b) An engineering system consists of 5 components connected in series, so, if one components fails, the system fails. The lifetimes (measured in years) of the 5 components, Y1,Y2,...,Y5, are all independent and identically distributed. (i) Suppose the lifetimes follow the standard uniform distribution U(0,1). Find the 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 6 months without failing. [10 marks] (ii) If, instead, the lifetimes follow an exponential distribution with mean θ, then Y(1) follows an exponential distribution with mean θ/5. Prove this result. Assuming that the only information available is a single observation on Y(1), find the most powerful test of size 0.05 for H0 : θ = θ1 versus H1 : θ = θ2, where θ1 < θ2. (Hint: the probability density function and cumulative distribution function for an exponential random variable with mean θ are f(y) = θ−1 exp(−y/θ), y > 0, and F(y) = 1−exp(−y/θ), y > 0, respectively.) [12 marks]

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