Let X1...., Xn be a random sample of size n from an infinite population and assume X1 d= a + bU2 with the constants a > 0 and b > 0 unknown and U a standard uniform distributed random variable given by FU (x) := P(U ≤ x) =    0 if x ≤ 0 x if 0 < x < 1 1 if x ≥ 1 1. Compute the cdf of the random variable X1. 2. Compute E(X1) and V ar(X1). 3. Give the method of moments estimators of the unknown parameters a and b. Explain how you construct these estimators! dLet X1..., Xn be a random sample of size n from an infinite population and assume X1a + 2Y with a > 0 unknown andY having an exponential distibution with parameter ) = 2. ThismeansFy (x) == P(Y < x) =1-e-2*, x > 0Since 0 = a is an unknown parameter and the random variables X; always satisfy X; > a apossible estimator for 0 = a could be given bymin{X1,..., Xn}.1.Compute the cdf of both the random variable X1 and the estimator 0.2.Compute the moment generating function E(es) for any s and the first momentE(0) and variance Var(0).Compute P(|0or both. Explain your answer!3.- a > e) for any e > 0. Is the estimator 0 unbiased or consistent4.Compute the mean squared error of the estimator 0 given byMSE(0) = E((@ – a)²).Solution

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Let X1...., Xn be a random sample of size n from an infinite population and assume X1 d= a + bU2 with the constants a > 0 and b > 0 unknown and U a standard uniform distributed random variable given by FU (x) := P(U ≤ x) =    0 if x ≤ 0 x if 0 < x < 1 1 if x ≥ 1 1. Compute the cdf of the random variable X1. 2. Compute E(X1) and V ar(X1). 3. Give the method of moments estimators of the unknown parameters a and b. Explain how you construct these estimators!

d
Let X1..., Xn be a random sample of size n from an infinite population and assume X1
a + 2Y with a > 0 unknown andY having an exponential distibution with parameter ) = 2. This
means
Fy (x) =
= P(Y < x) =1-e-2*, x > 0
Since 0 = a is an unknown parameter and the random variables X; always satisfy X; > a a
possible estimator for 0 = a could be given by
min{X1,..., Xn}.
1.
Compute the cdf of both the random variable X1 and the estimator 0.
2.
Compute the moment generating function E(es) for any s and the first moment
E(0) and variance Var(0).
Compute P(|0
or both. Explain your answer!
3.
- a > e) for any e > 0. Is the estimator 0 unbiased or consistent
4.
Compute the mean squared error of the estimator 0 given by
MSE(0) = E((@ – a)²).
Solution

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