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Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv)....
3. Suppose that we have two independent binomial random variablesXBinomial(n,pr) and Y~Binomial(m, p). You can assume...
3. Suppose that we have two independent binomial random variablesXBinomial(n,pr) and Y~Binomial(m, p). You can assume that the MLE's are p X/n and py- Y/m. (a) Find the MLE for p under the assumption that pr -Py (b) Find the LRT statistic T for testing (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 one degree of...
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...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
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...
3 (17') The random variable X obeys the distribution Binomial(n,p) with n=3, p=0.4. (a) Write Px(x),...
3 (17') The random variable X obeys the distribution Binomial(n,p) with n=3, p=0.4. (a) Write Px(x), the PMF of X. Be sure to write the value of Px(x) for all x from - to too. (b) Sketch the graph of the PMF Px [2] (c) Find E[X], the expected value of X. (d) Find Var[X], the variance of X.
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).
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 .
Let X and Y be two independent and identically distributed random variables that take only positive...
Let X and Y be two independent and identically distributed random variables that take only positive integer values. Their PMF is pX(n)=pY(n)=2−n for every n∈N , where N is the set of positive integers. Fix a t∈N . Find the probability P(min{X,Y}≤t) . Your answer should be a function of t . unanswered Find the probability P(X=Y) . unanswered Find the probability P(X>Y) . Hint: Use your answer to the previous part, and symmetry. unanswered Fix a positive integer k...
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...
3. Suppose that we have two independent binomial random variablesXBinomial(n,pr) and Y~Binomial(m, p). You can assume that the MLE's are p X/n and py- Y/m. (a) Find the MLE for p under the assumption that pr -Py (b) Find the LRT statistic T for testing (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 one degree of...
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...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
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...
3 (17') The random variable X obeys the distribution Binomial(n,p) with n=3, p=0.4. (a) Write Px(x), the PMF of X. Be sure to write the value of Px(x) for all x from - to too. (b) Sketch the graph of the PMF Px [2] (c) Find E[X], the expected value of X. (d) Find Var[X], the variance of 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...
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