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3. Suppose that we have two independent binomial random variablesXBinomial(n,pr) and Y~Binomial(m, p). You can assume...
Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv)....
Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv). You can assume that the MLE's are -X/n and p,-Y/m. (a) Find the MLE for p under the assumption that p (b) Find the LRT statistic T for testing p,-py HA:p.Ру vs. (c) Evaluate the value of this statistic if n 353, X 95, m -432, and Y 123. (d) Compare the answer from part (c) to a critical value from a x2 with...
5. we have two independent samples of n observations X1,X2, ,x, and Yi, ½, ,y, We...
5. we have two independent samples of n observations X1,X2, ,x, and Yi, ½, ,y, We want to test the hypothesis Ho M Hy versus the alternative Hi: Hr y (a) First, assume that the null hypothesis Ho is true and find the MLE for μ Ha-μυ (b) Then plug this estimate into the log likelihood along with the MLE's μ--x and μυ-D to calculate the LRT statistic (c) Is this likelihood ratio test equivalent to the test that rejects...
5. We have two independent samples of n observations X1, X2, .. . , Xn and...
5. We have two independent samples of n observations X1, X2, .. . , Xn and Yı, Y2,.. . , Yn We want to test the hvpothesis H 0 : μΧ-My versus the alternative H1 : μΧ * My (a) First, assume that the null hypothesis Ho is true and find the MLE for μ-Ac-My (b) Then plug this estimate into the log likelihood along with the MLE's μχ-x and My-- to calculate the LRT statistic (c) Is this likelihood...
5. We have two independent samples of n observations X1, X2,... , Xn and Yi, Y2,...,...
5. We have two independent samples of n observations X1, X2,... , Xn and Yi, Y2,..., Y, We want to test the hypothesis Ho : μ®-,ty versus the alternative H, : μ*-t ,ty. (a) First, assume that the null hypothesis Ho is true and find the MLE for μ-Ae-μΥ. (b) Then plug this estimate into the log likelihood along with the MLE's μΧ-x and My to calculate the LRT statistic. (c) Is this likelihood ratio test equivalent to the test...
Thank you so much! 5. We have two independent samples of n observations X1,Xy,.. . ,x,...
Thank you so much!
5. We have two independent samples of n observations X1,Xy,.. . ,x, and Y. Ya, . … We want to test the hypothesis Ho : μ,-μυ versus the alternative Hi : μ, μν. (a) First, assume that the null hypothesis Ho is true and find the MLE for μ (b) Then plug this estimate into the log likelihood along with the MLE μ'.. and 1,-j) to calculate the LRT statistic. (e) Is this likelihood ratio test...
Suppose X and Y are independent Binomial random variables, each with n=3 and p=9/10. a. Find...
Suppose X and Y are independent Binomial random variables, each with n=3 and p=9/10. a. Find the probability that X and Y are equal, i.e., find P(X=Y). b. Find the probability that X is strictly larger than Y, i.e., find P(X>Y). c. Find the probability that Y is strictly larger than X, i.e., find P(Y>X).
(2) Let Y be a binomial random variable with parameters n and p. Remember that We...
(2) Let Y be a binomial random variable with parameters n and p. Remember that We know that Y/n is an unbiased estimator of p. Now we want to estimate the variance of Y with n借)(1-n) (a) Find the expected value of this estimator (b) Find an unbiased estimator that is a simple modification of the proposed estimator
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3)...
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3) and Y is binomial(n = 3,p = 1/3). Find the moment-generating functions of the three random variables X, Y and X + Y . (You may look up the first two. The third follows from the first two and the behavior of moment-generating functions.) Now use the moment-generating function of X + Y to find the distribution of X + Y .
4. Let X,X,Bernoulli(p), and let Y Xi. Then we know that Y-Binomial(n.p) Consider the hypotheses Hop-Po...
4. Let X,X,Bernoulli(p), and let Y Xi. Then we know that Y-Binomial(n.p) Consider the hypotheses Hop-Po against Hip#po- a. c. For the particular case of po0.25 and n 5, fill in the table: 0 3 4 λ(y) P(Y - y) Find the (generalized) likelihood ratio test φ(y) of size α for testing Ho:p-po vs. H,: p d. po. Your test should be expressed in terms of y and α.
LetX,X2, , XnLLd. Bernoulli(p), and let Y-Σ,Xi. Then we know that Y-Binomial(n, p). 5. Consider t...
LetX,X2, , XnLLd. Bernoulli(p), and let Y-Σ,Xi. Then we know that Y-Binomial(n, p). 5. Consider the hypotheses Hop-po against HA:p#po- a. Find the likelihood function of p in terms of random variable Y, L(p). b. Construct the (generalized) likelihood ratio λ(v). Hint: what is pMLE?] C. (i) For the particular case of po 0.25 and n 5, fill in the table: 3 4 A(y) (ii) Rearrange the table in the order of increasing of values of 2, and compute cumulative...
Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv). You can assume that the MLE's are -X/n and p,-Y/m. (a) Find the MLE for p under the assumption that p (b) Find the LRT statistic T for testing p,-py HA:p.Ру vs. (c) Evaluate the value of this statistic if n 353, X 95, m -432, and Y 123. (d) Compare the answer from part (c) to a critical value from a x2 with...
5. we have two independent samples of n observations X1,X2, ,x, and Yi, ½, ,y, We want to test the hypothesis Ho M Hy versus the alternative Hi: Hr y (a) First, assume that the null hypothesis Ho is true and find the MLE for μ Ha-μυ (b) Then plug this estimate into the log likelihood along with the MLE's μ--x and μυ-D to calculate the LRT statistic (c) Is this likelihood ratio test equivalent to the test that rejects...
5. We have two independent samples of n observations X1, X2, .. . , Xn and Yı, Y2,.. . , Yn We want to test the hvpothesis H 0 : μΧ-My versus the alternative H1 : μΧ * My (a) First, assume that the null hypothesis Ho is true and find the MLE for μ-Ac-My (b) Then plug this estimate into the log likelihood along with the MLE's μχ-x and My-- to calculate the LRT statistic (c) Is this likelihood...
5. We have two independent samples of n observations X1, X2,... , Xn and Yi, Y2,..., Y, We want to test the hypothesis Ho : μ®-,ty versus the alternative H, : μ*-t ,ty. (a) First, assume that the null hypothesis Ho is true and find the MLE for μ-Ae-μΥ. (b) Then plug this estimate into the log likelihood along with the MLE's μΧ-x and My to calculate the LRT statistic. (c) Is this likelihood ratio test equivalent to the test...
Thank you so much!
5. We have two independent samples of n observations X1,Xy,.. . ,x, and Y. Ya, . … We want to test the hypothesis Ho : μ,-μυ versus the alternative Hi : μ, μν. (a) First, assume that the null hypothesis Ho is true and find the MLE for μ (b) Then plug this estimate into the log likelihood along with the MLE μ'.. and 1,-j) to calculate the LRT statistic. (e) Is this likelihood ratio test...
(2) Let Y be a binomial random variable with parameters n and p. Remember that We know that Y/n is an unbiased estimator of p. Now we want to estimate the variance of Y with n借)(1-n) (a) Find the expected value of this estimator (b) Find an unbiased estimator that is a simple modification of the proposed estimator
4. Let X,X,Bernoulli(p), and let Y Xi. Then we know that Y-Binomial(n.p) Consider the hypotheses Hop-Po against Hip#po- a. c. For the particular case of po0.25 and n 5, fill in the table: 0 3 4 λ(y) P(Y - y) Find the (generalized) likelihood ratio test φ(y) of size α for testing Ho:p-po vs. H,: p d. po. Your test should be expressed in terms of y and α.
LetX,X2, , XnLLd. Bernoulli(p), and let Y-Σ,Xi. Then we know that Y-Binomial(n, p). 5. Consider the hypotheses Hop-po against HA:p#po- a. Find the likelihood function of p in terms of random variable Y, L(p). b. Construct the (generalized) likelihood ratio λ(v). Hint: what is pMLE?] C. (i) For the particular case of po 0.25 and n 5, fill in the table: 3 4 A(y) (ii) Rearrange the table in the order of increasing of values of 2, and compute cumulative...
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