Help PLEASE! 2. Find M.L.E for the parameter 0 There are 3 observations, X1 = 1,X2...
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3. Find M.M.E for the parameter 0 There are 3 observations, X1 3(1-0) = 1,X2 2,X3 = 1 20 P(X1 1) = P(X1 2) = 3 - e 3-0
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4. Find M.L.E for the parameter 0 There are 3 observations, X, = 0.1, X2 = 0.5,X3 = 0.8 2 2x f(x) = -22, 0<x<e
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5. Find M.M.E for the parameter 0 There are 3 observations, X1 = 0.1, X2 0.5, X3 = 0.8 2 2x 0x<e f(x) = 02
X1 , X2 , X3 ~ exponential(1) then find P(max(X1 , X2 , X3)<2 | X1 + X2 + X3 = 3) = ?
3. Suppose that X1, X2, X3 be i.i.d. random variables with P(Xi 0) 2/5 and P(X 1) 3/5. Find the MGFof X, + X2 + X 3.
3. Suppose that X1, X2, X3 be i.i.d. random variables with P(Xi 0) 2/5 and P(X 1) 3/5. Find the MGFof X, + X2 + X 3.
1. Suppose that X1, X2, and X3 E(X1) = 0, E(X2) = 1, E(X3) = 1, Var(X1) = 1, Var(X2) = 2, Var(X3) = 3, Cov(X1, X2) = -1, Cov(X2, X3) = 1, where X1 and X3 are independent. a.) Find the covariance cov(X1 + X2, X1 - X3). b.) Define U = 2X1 - X2 + X3. Find the mean and variance of U.
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Question 6 [2 marks] Let X1, X2, ..., X, be a random sample from the Poisson distribution with mean e. a. Express the VAR,(Xi) as a function o2 = g(e). b. b. Find the M.L.E. of g(0) and show that it is unbiased.
3. Given X1 ~ N(0,1), X2 ~ N(20,1) with unknown parameter 0. X1 and X2 are independent. Derive the most powerful a-level test for Ho : 0 = 0 vs. H1 : 0 = 1 using both X1 and X2. Give an implementable form of this test. (Note that our sample X1 and X2 have different distributions now, but you can still write out the likelihood function for Xį and X2 jointly, and then use the N-P lemma as usual.)
For the data x1 = -1, x2 =
-3, x3 = -2, x4 =
1, x5 = 0,
find ∑
(xi2).
A parameter ξ is obtained from n independent observations (x1, x2, ...). Construct a posterior probability density function p(ξ|x1,x2,...) for ξ.