Exercise 4. (20 points) Let X1...., Xn be a random sample of size n from an infinite population and assume X1 d= a + 2Y with a > 0 unknown and Y having an exponential distibution with parameter = 2. This means FY (x) = P(Y x) = 1 e 2x , x > 0 Since = a is an unknown parameter and the random variables Xi always satisfy Xi > a a possible estimator for = a could be given by b = min{X1, ..., Xn}.

Exercise 4. (20 points) Let X1..., Xn be a random sample of size n from an infinite population and assume X a+2Y with a > 0 unknown and Y having an exponential distibution with parameter 1 = 2. This means Fy(x) = P(Y 0 Since 0 = a is an unknown parameter and the random variables X; always satisfy Xi > a a possible estimator for 0 = a could be given by @ = min{X1, ..., Xn} 1. (5 points) Compute the cdf of both the random variable X, and the estimator . 2. (5 points) Compute the moment generating function E(es) for any s and the first moment El) and variance Var(@). 3. (5 points) Compute P( - a > ) for any e > 0. Is the estimator unbiased or consistent or both. Explain your answer! 4. (5 points) Compute the mean squared error of the estimator @ given by MSE(0) = E(6 a)^. Exercise 4. (20 points) Let X1..., Xn be a random sample of size n from an infinite population and assume X a+2Y with a > 0 unknown and Y having an exponential distibution with parameter 1 = 2. This means Fy(x) = P(Y 0 Since 0 = a is an unknown parameter and the random variables X; always satisfy Xi > a a possible estimator for 0 = a could be given by @ = min{X1, ..., Xn} 1. (5 points) Compute the cdf of both the random variable X, and the estimator . 2. (5 points) Compute the moment generating function E(es) for any s and the first moment El) and variance Var(@). 3. (5 points) Compute P( - a > ) for any e > 0. Is the estimator unbiased or consistent or both. Explain your answer! 4. (5 points) Compute the mean squared error of the estimator @ given by MSE(0) = E(6 a)^