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Math 562 Homework 1 August 29, 2006 Dr. Ron Sahoo He who labors diligently need never despair; for all things are accomplished by diligence and labor. — Menander of Athens Direction: This homework worths 60 points and is due on September 14, 2006. In order to receive full credit, answer each problem completely and must show all work. 1. Seven observations are drawn from a population with an unknown con- tinuous distribution. What is the probability that the least and the greatest observations bracket the median? 2. If the random variable X has the density function 2 (1 − x) for 0 ≤ x ≤ 1 f (x) = 0 otherwise, what is the probability that the larger of 2 independent observations of X will exceed 12 ? 3. Let X1 , X2 , X3 , X4 , X5 be a random sample from the uniform distribution on the interval (0, θ), where θ is unknown, and let Xmax denote the largest observation. For what value of the constant k, the expected value of the random variable kXmax is equal to θ? 4. Five observations have been drawn independently and at random from a continuous distribution. What is the probability that the next observation will be less than all of the first 5? 5. Let X and Y be two independent random variables with identical probability density function given by 2 3θx3 for 0 ≤ x ≤ θ f (x) = 0 elsewhere, for some θ > 0. What is the probability density function of W = min{X, Y }? 6. Let X1 , X2 , X3 be a random sample of size 3 from a standard normal dis- tribution. Find the sampling distribution of the statistics √X1 2+X2 +X 2 3 2 and X1 +X2 +X3 √X1 −X 2 2 −X3 2 2 . X1 +X2 +X3 7. Suppose X1 , X2 , ...., Xn is a random sample from a normal distribution with Pn Pn 2 mean µ and variance σ 2 . If X̄ = n1 i=1 Xi and Σ2 = n1 i=1 Xi − X̄ , and Xn+1 is an additional observation, what is the value of m so that the statistics m(X̄−Xn+1 ) Σ has a t-distribution. 8. Let X1 , X2 , X3 , X4 be a random sample of size 4 from a standard normal population. Find the distribution of the statistic √X1 +X 2 4 2 . X2 +X3 9. Let X1 , X2 , ..., Xn be a random sample from a normal distribution with Pn 2 mean µ and variance σ 2 . What is the variance of Σ2 = n1 i=1 Xi − X ? 10. A random sample X1 , X2 , ..., Xn of size n is selected from a normal popu- lation with mean µ and standard deviation 1. Later an additional independent observation Xn+1 is obtained from the same population. What is the distribu- Pn tion of the statistic (Xn+1 − µ)2 + i=1 (Xi − X)2 , where X denote the sample mean? 11. Suppose Xj = Zj − Zj−1 , where j = 1, 2, ..., n and Z0 , Z1 , ..., Zn are independent and identically distributed with common variance σ 2 . What is the Pn variance of the random variable n1 j=1 Xj ? 12. Let X1 , X2 , ..., Xn and Y1 , Y2 , ..., Yn be two random sample from the in- dependent normal distributions with V ar[Xi ] = σ 2 and V ar[Yi ] = 2σ 2 , for Pn 2 Pn 2 i = 1, 2, ..., n and σ 2 > 0. If U = i=1 Xi − X and V = i=1 Yi − Y , then what is the sampling distribution of the statistic 2U2σ+V2 ? 13. Let X1 , X2 , ..., X9 be a random sample of size 9 from a distribution with a probability density function θ xθ−1 if 0<x<1 f (x; θ) = 0 otherwise, where 0 < θ is a parameter. Using the moment method find an estimator for the parameter θ. 14. Let X1 , X2 , ..., X7 be a random sample of size 7 from a distribution with a probability density function θ2

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