Consider a random sample X1,..., X, with n > 3 from the exponential distribution rith parameter A > 0 and density f(x; A)- I>0. Consider the following two estimators of A:
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- For a statistic to be a good estimator of a parameter, two properties it must satisfy are unbiasedness and minimum variance. Consider a sample of three observations X₁, X₂, X, where X, ~ Exp (0). That is, a sample of size 3 is taken from a population 2 3 following the exponential distribution with density function given by f(x) = 1e % if x > 0 0, Otherwise. Five possible estimators of are â‚ =X₁, Ô₂ = ¹/(X₁ + X₂), Ô‚ = =— (X₁ + 2X₂), Ô¸ = X, and Ô¸ = ¹⁄ (X₂ + X). 2 [Hint: Use the fact that for variable X we have E(X)= 0 and E(X²)=20² and Var(X) = 0². (a) Show that the five estimators given above are unbiased for 0. (b) Find the variances of each of the five estimators. (c) Which estimator will you choose for 0. Why?Let X. X , be a random sample of siren from a normal distribution with population mean 0 and population variance a>Ois unknown, that is N(0, ). Suppose that we define the following plvotal quantity (n-1)s where & and S' are the sample mean and the sample variance, respectively. Use the plvotal quantity (Q) to derive the (1- a) 100% confidence interval (C) for the unknown parameter (o) if the two critical values are q--n, a) and (n-1) (A) CI = (n8+(n-1)51 + n-1) s (8) CI - (C) CI - + (n-1) s (D) CI= O D O A O c O BLet X₁,..., Xn be a random sample from a distribution with one of two pdfs. If 0 = 1, then f(x;0=1) = 1 (0 < x < 1). If 0 = 2, then f(x;0= 2) = 2x1 (0 < x < 1). (a) Give a general form of the MLE of 0. (b) You are given a sample data of 3 values: 0.4, 0.5, 0.8. Find the MLE estimate of 0 based on the data.
- Consider a random sample X1,..., X, with n > 3 from the exponential distribution with parameter A > 0 and density 1 f(x; A) = e/A I> 0. Consider the following two estimators of A: 3 Ex,, X =nX1). = i=1Suppose X is a random variable defined as final cost of project estimated cost of project which has a PDF as follows: [0 xa a. Determine the value of a. b. What is the probability that the final cost of a project will exceed its estimated cost by 25%? c. Determine the mean value and standard deviation of X.3. Let X1, .… , Xn be a random sample. Find the mle of 0 for the cases that the population pdf/pmf is, f(r;:0) = 0, %3D otherwise
- 2. The random variable X is the amount of fructose in apples (in ounces) grown at a local orchard and is represented by the following cdf: 2(x +), 1Let X be a normal random variable with mean 85 and a variance of 25 (i.e., X ∼ N (85, 25)). step1: Let M = aX + b for some constants a, b not equal to 0. Write an expression for the pdf of M . step 2: Suppose that X represents an approximate distribution of the final scores in a certain math course (ignore the fact that this approximation can technically have scores that are greater than 100 or less than 0). If the teacher were to curve the scores, it means he would determine a function to apply to the scores to achieve a desired distribution (assume an affine function in this case, as in the previous part). Suppose he wants the scores to be normally distributed with mean 80 and a standard deviation of 4. What should he choose for a and b? What students would see their score lowered, and what students would see their score increased?Let L be the random variable for the length of time, in years, that a person will remember an actuarial statistic. For a certain popula- tion, L is exponentially distributed with mean 1/Y, where Y has a gamma distribution with a = 4.5 and 3 = 4. Find the variance of L.Let X1, X2, X3, be a r. s. from normal distribution with mean 0 and variance 1, let the prior distribution of 0 is normal with mean zeroand variance 1. Let L(6,0) = (ô - 0). Find the Bayes estimator of 0Let Y < Y, < Y3 be the order statistic of a random sample of size 3 from the uniform distribution having pdff (x;0) = 1/0,06. Find the median of an exponential random variable X with parameter 1 = 4. (The median of X is m iff P(X < m) = 0.5.) Does it differ from E(X)? Try to explain.Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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