Homework Answers
According to invariance property of MLE, we can say that
MLE of 
is 
where 
is the MLE of 
.
In this case,
MLE of
will be 
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2.3 Let X be a r.v. describing the lifetime of a certain equipment, and suppose that...
3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0...
3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0>0, and let be its expected lifetime. On the basis of the random sample Xi,..X from this distribution, derive the MP test for testing the hypothesis Ho:to against the alternative HA:-(>o) at level of significance o
3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0>0, and let be its expected...
x, R(x), is defined as the probability that X>x; ie, R(x)-P(X> x). Now, suppose that X...
x, R(x), is defined as the probability that X>x; ie, R(x)-P(X> x). Now, suppose that X has the Negative Exponential p.d.f, Then (ii) Use Theorem 1 in order to determine the MLE of R(x; θ), on the basis of a random sample X1,…, X from the underlying p.d.f. Theorem 1 Let θ-θ(x) be the MLE of θ on the basis of the observed values x1, , xn of the random sample X1, , X, from the pdf f(", θ), θ...
Practice Exam Questions 2 Let X be a r.v. with density function x 2 1 a. Determine the distributi...
Practice Exam Questions 2 Let X be a r.v. with density function x 2 1 a. Determine the distribution function of X, i.e. F(x). Find E(X) and V(X) b. Find the MLE estimator of θ constructed from a sample Xi,Xn c. Is the estimator find in (b) biased?
Practice Exam Questions 2 Let X be a r.v. with density function x 2 1 a. Determine the distribution function of X, i.e. F(x). Find E(X) and V(X) b. Find the MLE...
2. A certain type of electronic component has a lifetime X (in hours) with probability density...
2. A certain type of electronic component has a lifetime X (in hours) with probability density function given by otherwise. where θ 0. Let X1, . . . , Xn denote a simple random sample of n such electrical components. . Find an expression for the MLE of θ as a function of X1 Denote this MLE by θ ·Determine the expected value and variance of θ. » What is the MLE for the variance of X? Show that θ...
Let Xi , X2,. … X, denote a random sample of size n > 1 from a distribution with pdf f(x:0)--x'e®, x > 0 and θ > 0. a. Find the MLE for 0 b. Is the MLE unbiased? Show your steps. c. Find...
Let Xi , X2,. … X, denote a random sample of size n > 1 from a distribution with pdf f(x:0)--x'e®, x > 0 and θ > 0. a. Find the MLE for 0 b. Is the MLE unbiased? Show your steps. c. Find a complete sufficient statistic for 0. d. Find the UMVUE for θ. Make sure you indicate how you know it is the UMVUE.
Let Xi , X2,. … X, denote a random sample of size n...
explan the answer 10: A certain continuous distribution has cumulative distribution function (CDF) given by F(r)...
explan the answer
10: A certain continuous distribution has cumulative distribution function (CDF) given by F(r) 0, <0 where θ is an unknown parameter, θ > 0. (i) Find (a) the p.d.f., (b) the mean and (e) the variance of this distribution. (ii) Suppose that X (Xi, X2, Xn) is a random sample from this distribu- tion and let Y max(Xi, XXn). Find the CDF and p.d.f. of Y. Hence find the value of a for which EloY)
Suppose the average lifetime of a certain type of car battery is known to be 60...
Suppose the average lifetime of a certain type of car battery is known to be 60 months. Consider conducting a two-sided test on it based on a sample of size 25 from a normal distribution with a population standard deviation of 4 months. a) If the true average lifetime is 62 months and a=0.01, what is the probability of a type II error? b) What is the required sample size to satisfy and the type II error probability of b(62)...
Exercise 4.8: Suppose that X1, X2,..., Xn is a random sample of observations on a r.v. X, which takes values only in the range (0, 1). Under the null hypothesis Ho, the distribution of X is uniform o...
Exercise 4.8: Suppose that X1, X2,..., Xn is a random sample of observations on a r.v. X, which takes values only in the range (0, 1). Under the null hypothesis Ho, the distribution of X is uniform on (0, 1), whereas under an alternative hypothesis, њ, the distribution is the truncated exponential with p.d.f. 0e8 where 6 is unknown. Show that there is a UMP test of Ho vs Hi and find, roximately, the critical region for such a test...
4. Let Xi,..., Xn be a random sample with density 303 for 0 < θ < x NOTE: We have previously foun...
4. Let Xi,..., Xn be a random sample with density 303 for 0 < θ < x NOTE: We have previously found that θMLE-X(1) and that FX(1) (x)-1-(!)3m (a) Using the probability integral transform method, find a pivot for 0 based on the MLE. (b) Use the pivot found in (a) to get an ezact 100(1-a)% C.1. for θ (c) Find an approximate 100(1-a)% C.1. for θ based on our result for the MLE. (d) Suppose that we get n...
Let X be a random variable with p.d.f. f(x) = θx^(θ−1) , for 0 < x...
Let X be a random variable with p.d.f. f(x) = θx^(θ−1) , for 0 < x < 1. Let X1, ..., Xn denote a random sample of size n from this distribution. (a) Find E(X) [2] (b) Find the method of moment estimator of θ [2] (c) Find the maximum likelihood estimator of θ [3] (d) Use the following set of observations to obtain estimates of the method of moment and maximum likelihood estimators of θ. [1 each] 0.0256, 0.3051,...
3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0>0, and let be its expected lifetime. On the basis of the random sample Xi,..X from this distribution, derive the MP test for testing the hypothesis Ho:to against the alternative HA:-(>o) at level of significance o
3. (10 points) The life of an electronic equipment is a r.v. X whose p.d.f is f(z;θ)-Be 4xz > 0,0>0, and let be its expected...
x, R(x), is defined as the probability that X>x; ie, R(x)-P(X> x). Now, suppose that X has the Negative Exponential p.d.f, Then (ii) Use Theorem 1 in order to determine the MLE of R(x; θ), on the basis of a random sample X1,…, X from the underlying p.d.f. Theorem 1 Let θ-θ(x) be the MLE of θ on the basis of the observed values x1, , xn of the random sample X1, , X, from the pdf f(", θ), θ...
Practice Exam Questions 2 Let X be a r.v. with density function x 2 1 a. Determine the distribution function of X, i.e. F(x). Find E(X) and V(X) b. Find the MLE estimator of θ constructed from a sample Xi,Xn c. Is the estimator find in (b) biased?
Practice Exam Questions 2 Let X be a r.v. with density function x 2 1 a. Determine the distribution function of X, i.e. F(x). Find E(X) and V(X) b. Find the MLE...
2. A certain type of electronic component has a lifetime X (in hours) with probability density function given by otherwise. where θ 0. Let X1, . . . , Xn denote a simple random sample of n such electrical components. . Find an expression for the MLE of θ as a function of X1 Denote this MLE by θ ·Determine the expected value and variance of θ. » What is the MLE for the variance of X? Show that θ...
Let Xi , X2,. … X, denote a random sample of size n > 1 from a distribution with pdf f(x:0)--x'e®, x > 0 and θ > 0. a. Find the MLE for 0 b. Is the MLE unbiased? Show your steps. c. Find a complete sufficient statistic for 0. d. Find the UMVUE for θ. Make sure you indicate how you know it is the UMVUE.
Let Xi , X2,. … X, denote a random sample of size n...
explan the answer
10: A certain continuous distribution has cumulative distribution function (CDF) given by F(r) 0, <0 where θ is an unknown parameter, θ > 0. (i) Find (a) the p.d.f., (b) the mean and (e) the variance of this distribution. (ii) Suppose that X (Xi, X2, Xn) is a random sample from this distribu- tion and let Y max(Xi, XXn). Find the CDF and p.d.f. of Y. Hence find the value of a for which EloY)
Exercise 4.8: Suppose that X1, X2,..., Xn is a random sample of observations on a r.v. X, which takes values only in the range (0, 1). Under the null hypothesis Ho, the distribution of X is uniform on (0, 1), whereas under an alternative hypothesis, њ, the distribution is the truncated exponential with p.d.f. 0e8 where 6 is unknown. Show that there is a UMP test of Ho vs Hi and find, roximately, the critical region for such a test...
4. Let Xi,..., Xn be a random sample with density 303 for 0 < θ < x NOTE: We have previously found that θMLE-X(1) and that FX(1) (x)-1-(!)3m (a) Using the probability integral transform method, find a pivot for 0 based on the MLE. (b) Use the pivot found in (a) to get an ezact 100(1-a)% C.1. for θ (c) Find an approximate 100(1-a)% C.1. for θ based on our result for the MLE. (d) Suppose that we get n...
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