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1. X,,x2,..., X, is a random sample from a Poisson (0) distribution with probability mass function...
(a)Suppose X ∼ Poisson(λ) and Y ∼ Poisson(γ) are independent, prove that X + Y ∼...
(a)Suppose X ∼ Poisson(λ) and Y ∼ Poisson(γ) are independent, prove that X + Y ∼ Poisson(λ + γ). (b)Let X1, . . . , Xn be an iid random sample from Poisson(λ), provide a sufficient statistic for λ and justify your answer. (c)Under the setting of part (b), show λb = 1 n Pn i=1 Xi is consistent estimator of λ. (d)Use the Central Limit Theorem to find an asymptotic normal distribution for λb defined in part (c), justify...
Let X1,X2,,X be a random sample from a distribution function f(x,8) = θ"(1-8)1-r for x =...
Let X1,X2,,X be a random sample from a distribution function f(x,8) = θ"(1-8)1-r for x = 0,1 (a) Show that Y = Σ.1X, is a sufficient statistic for θ. (i) Find a function of Y that is an unbiased estimate for θ (ii) Hence, explain why this function is the minimum variance unbiased estimator(MVUE) for θ (c) Is1-the MVUE for Please explain.
Consider a random sample of size n from the distribution with pdf (In )* f(x; 0)...
Consider a random sample of size n from the distribution with pdf (In )* f(x; 0) = { 0.c! -, 10, =0,1,... otherwise where 0 > 0. (a) (10 pts) Find a complete sufficient statistic for 0. (b) (10 pts) Using Lehmann-Scheffe theorem, find the UMVUE of Ine. You may need the identity c=
2-3. Let ?>0 and ?? R. Let X1,X2, distribution with probability density function , Xn be...
2-3. Let ?>0 and ?? R. Let X1,X2, distribution with probability density function , Xn be a random sample from the zero otherwise suppose ? is known. ( Homework #8 ): W-X-5 has an Exponential ( 2. Recall --)-Gamma ( -1,0--) distribution. a) Find a sufficient statistic Y-u(X1, X2, , Xn) for ? b) Suggest a confidence interval for ? with (1-?) 100% confidence level. "Flint": Use ?(X,-8) ? w, c) Suppose n-4, ?-2, and X1-215, X2-2.55, X3-210, X4-2.20. i-1...
Please let me know how to solve 7.6.5. 6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with p...
Please let me know how to solve 7.6.5.
6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with parameter θ > 0. (a) Find the MVUE of P(X < 1)-(1 +0)c". Hint: Let u(x)-1, where Y = Σ1Xi. 1, zero elsewhere, and find Elu(Xi)|Y = y, xỉ (b) Express the MVUE as a function of the mle of θ. (c) Determine the asymptotic distribution of the mle of θ (d) Obtain the mle of P(X...
4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with...
4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with probability density function S(*;ð) - ma t?e-vor x>0, zero otherwise. Recall: W=vX has Gamma( a -6, 0-ta) distribution. Y=ZVX; = Z W; has a Gamma ( a =6n, = ta) distribution. i=1 E(Xk) - I( 2k+6) 120 ok k>-3. 42 S. A method of moments estimator of 8 is 42.n 8 = h) Suggest a confidence interval for 8 with (1 - 0) 100%...
3. Suppose that X (X...,X) is a random sample from a uniform distribution of the interval...
3. Suppose that X (X...,X) is a random sample from a uniform distribution of the interval [0,0], where the value of ? is unknown, and it is desired to test the hypotheses H: 0>2 [5] (a) Show that the uniform family f(x;0)-(1/0)1 om(r) : ? > 0 maxi-isnXi. has a monotone likelihood ratio in the statistic T(X)- X. whereX (n) [5] (b) Find a uniformly most powerful (UMP) test of level ? for testing Ho versus HI
A probability distribution function for a random variable X has the form Fx(x) = A{1 -...
A probability distribution function for a random variable X has the form Fx(x) = A{1 - exp[-(x - 1)]}, 1<x< 10, -00<x<1 (a) For what value of A is this a valid probability distribution function? (b) Find the probability density function and sketch it. (c) Use the density function to find the probability that the random variable is in the range 2 < X <3. Check your answer using the distribution function. (d) Find the probability that the random variable...
The Poisson distribution with parameter λ has the mass function defined by p(x) = λ x...
The Poisson distribution with parameter λ has the mass function defined by p(x) = λ x e −λ/x! if x is a nonnegative integer (and 0 otherwise). Find the probability it assigns to each of the following sets: a. [0, 2) b. (−∞,1] c. (−∞,1.5] d. (−∞, 2) e. (−∞,2] f. (0.5, ∞) g. {0, 1, 2} Find the CDF of the uniform distribution on (0,1).
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the...
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0 unbiased estimator of e. estimator eCYis an (c) Get the lower bound for the variance of the unbiased estimator found in (b)
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0...
Let X1,X2,,X be a random sample from a distribution function f(x,8) = θ"(1-8)1-r for x = 0,1 (a) Show that Y = Σ.1X, is a sufficient statistic for θ. (i) Find a function of Y that is an unbiased estimate for θ (ii) Hence, explain why this function is the minimum variance unbiased estimator(MVUE) for θ (c) Is1-the MVUE for Please explain.
Consider a random sample of size n from the distribution with pdf (In )* f(x; 0) = { 0.c! -, 10, =0,1,... otherwise where 0 > 0. (a) (10 pts) Find a complete sufficient statistic for 0. (b) (10 pts) Using Lehmann-Scheffe theorem, find the UMVUE of Ine. You may need the identity c=
2-3. Let ?>0 and ?? R. Let X1,X2, distribution with probability density function , Xn be a random sample from the zero otherwise suppose ? is known. ( Homework #8 ): W-X-5 has an Exponential ( 2. Recall --)-Gamma ( -1,0--) distribution. a) Find a sufficient statistic Y-u(X1, X2, , Xn) for ? b) Suggest a confidence interval for ? with (1-?) 100% confidence level. "Flint": Use ?(X,-8) ? w, c) Suppose n-4, ?-2, and X1-215, X2-2.55, X3-210, X4-2.20. i-1...
Please let me know how to solve 7.6.5.
6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with parameter θ > 0. (a) Find the MVUE of P(X < 1)-(1 +0)c". Hint: Let u(x)-1, where Y = Σ1Xi. 1, zero elsewhere, and find Elu(Xi)|Y = y, xỉ (b) Express the MVUE as a function of the mle of θ. (c) Determine the asymptotic distribution of the mle of θ (d) Obtain the mle of P(X...
4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with probability density function S(*;ð) - ma t?e-vor x>0, zero otherwise. Recall: W=vX has Gamma( a -6, 0-ta) distribution. Y=ZVX; = Z W; has a Gamma ( a =6n, = ta) distribution. i=1 E(Xk) - I( 2k+6) 120 ok k>-3. 42 S. A method of moments estimator of 8 is 42.n 8 = h) Suggest a confidence interval for 8 with (1 - 0) 100%...
3. Suppose that X (X...,X) is a random sample from a uniform distribution of the interval [0,0], where the value of ? is unknown, and it is desired to test the hypotheses H: 0>2 [5] (a) Show that the uniform family f(x;0)-(1/0)1 om(r) : ? > 0 maxi-isnXi. has a monotone likelihood ratio in the statistic T(X)- X. whereX (n) [5] (b) Find a uniformly most powerful (UMP) test of level ? for testing Ho versus HI
A probability distribution function for a random variable X has the form Fx(x) = A{1 - exp[-(x - 1)]}, 1<x< 10, -00<x<1 (a) For what value of A is this a valid probability distribution function? (b) Find the probability density function and sketch it. (c) Use the density function to find the probability that the random variable is in the range 2 < X <3. Check your answer using the distribution function. (d) Find the probability that the random variable...
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0 unbiased estimator of e. estimator eCYis an (c) Get the lower bound for the variance of the unbiased estimator found in (b)
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0...
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