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Problem 6 늪). Suppose X ~ Uniform(0, 1), and given X = x, Y ~ Exponential(λ = 하 a. Find the lin...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
Let X be uniform on [0, 1], and let Y be exponential with rate λ, so...
Let X be uniform on [0, 1], and let Y be exponential with rate λ, so that P(Y ≥ t) = e ^(−λt ) t ≥ 0 and 1 if t < 0 Assume that X and Y are independent, and define W = X + 2Y . a) For any w ≥ 0 and x ∈ [0, 1], compute P(W ≥ w|X = x) b) By undoing the conditioning on X, use the result from part (a) to compute...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is expo...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...
Problem 6: Nonlinear Channel Suppose a transmitter sends a signal X over a channel, where X is exponentially distributed with mean 1. The channel non-linearly distorts the transmitted signal, such th...
Problem 6: Nonlinear Channel Suppose a transmitter sends a signal X over a channel, where X is exponentially distributed with mean 1. The channel non-linearly distorts the transmitted signal, such that the signal at the receiver is given by Y-1-e-Ax, where λ > 0 is a constant. The receiver then estimates X using a linear estimator X based on Y which minimizes the mean squared error between X and X. Find this linear estimator
Problem 6: Nonlinear Channel Suppose a...
Problem 6: Nonlinear Channel Suppose a transmitter sends a signal X over a channel, where X is exponentially distributed with mean 1. The channel non-linearly distorts the transmitted signal, such th...
Problem 6: Nonlinear Channel Suppose a transmitter sends a signal X over a channel, where X is exponentially distributed with mean 1. The channel non-linearly distorts the transmitted signal, such that the signal at the receiver is given by Y-1-e-Ax, where λ > 0 is a constant. The receiver then estimates X using a linear estimator X based on Y which minimizes the mean squared error between X and X. Find this linear estimator
Problem 6: Nonlinear Channel Suppose a...
Problem 3 Uniform Order Stats as Estimators Suppose that X,.., X,, ~ Unif(0, 0) are independent....
Problem 3 Uniform Order Stats as Estimators Suppose that X,.., X,, ~ Unif(0, 0) are independent. Consider 0, = 2X Part 1 Find MSE() Part 2 Consider an estimator of the form 02 = cX(n). Find c such that B(02) = 0. Then, find MSE(02). Which estimator is better? Part 3 was that 0 E (0,0) we can show that there is no unbiased In class we briefly discussed that a property we wanted for 0 If we restrict to...
11.3) Bayesian Parameter Estimation. Suppose Λ is a random parameter with prior given by the Gamma...
11.3) Bayesian Parameter Estimation. Suppose Λ is a random
parameter with prior given by the Gamma density
7(a) = CM2-1/4 2 0}, where a is a known positive real number, and I is the Gamma function defined by the integral ['(x) = ( +12'dt, for x > 0 Jo Our observation Yis Poisson with rate A, i.e., p(y) = P({Y = y}|{A =2}) = - - ale-2 - y = 0, 1,2,.... O y! (a) Find the MAP estimate of...
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ ...
difficult……
2and4 thanks
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ ...
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ. (b) Find an...
Ouestion 7 (10 points)Suppose Y..... y denote a random sample of size n from an exponential...
Ouestion 7 (10 points)Suppose Y..... y denote a random sample of
size n from an exponential distribu-| tion with mean 9.a) (5
points)Find the bias and MSE of the estimator B1 = nY().b) (3
points)Consider another estimator B, =Y. Find the efficiency of 6,
relative to 62.e) (7 points)Prove that 2 is a pivotal quantity and
find a 95% confidence interval for 8.
Question 7 (10 points) Suppose Y1, ..., Yn denote a random sample of size n from an...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...
Problem 6: Nonlinear Channel Suppose a transmitter sends a signal X over a channel, where X is exponentially distributed with mean 1. The channel non-linearly distorts the transmitted signal, such that the signal at the receiver is given by Y-1-e-Ax, where λ > 0 is a constant. The receiver then estimates X using a linear estimator X based on Y which minimizes the mean squared error between X and X. Find this linear estimator
Problem 6: Nonlinear Channel Suppose a...
Problem 6: Nonlinear Channel Suppose a transmitter sends a signal X over a channel, where X is exponentially distributed with mean 1. The channel non-linearly distorts the transmitted signal, such that the signal at the receiver is given by Y-1-e-Ax, where λ > 0 is a constant. The receiver then estimates X using a linear estimator X based on Y which minimizes the mean squared error between X and X. Find this linear estimator
Problem 6: Nonlinear Channel Suppose a...
Problem 3 Uniform Order Stats as Estimators Suppose that X,.., X,, ~ Unif(0, 0) are independent. Consider 0, = 2X Part 1 Find MSE() Part 2 Consider an estimator of the form 02 = cX(n). Find c such that B(02) = 0. Then, find MSE(02). Which estimator is better? Part 3 was that 0 E (0,0) we can show that there is no unbiased In class we briefly discussed that a property we wanted for 0 If we restrict to...
11.3) Bayesian Parameter Estimation. Suppose Λ is a random
parameter with prior given by the Gamma density
7(a) = CM2-1/4 2 0}, where a is a known positive real number, and I is the Gamma function defined by the integral ['(x) = ( +12'dt, for x > 0 Jo Our observation Yis Poisson with rate A, i.e., p(y) = P({Y = y}|{A =2}) = - - ale-2 - y = 0, 1,2,.... O y! (a) Find the MAP estimate of...
difficult……
2and4 thanks
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ. (b) Find an...
Ouestion 7 (10 points)Suppose Y..... y denote a random sample of
size n from an exponential distribu-| tion with mean 9.a) (5
points)Find the bias and MSE of the estimator B1 = nY().b) (3
points)Consider another estimator B, =Y. Find the efficiency of 6,
relative to 62.e) (7 points)Prove that 2 is a pivotal quantity and
find a 95% confidence interval for 8.
Question 7 (10 points) Suppose Y1, ..., Yn denote a random sample of size n from an...
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