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Properties of the expectation
Problem 1. Consider the five-parameter model given by the mixture of two...
3. Let X1, . . . , Xn be iid random variables with mean μ and...
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
Problem 5 Information Problem 6: 10 points A) Find moments of first and second order and...
Problem 5 Information
Problem 6: 10 points A) Find moments of first and second order and the variance for variable M in Problem 5 (B) Find expectation and variance for variable N in Problem 5. Problem 5: 10 points Consider n independent variables, {X1, X2,... , Xn} uniformly distributed over the unit interval, (0,1) Introduce two new random variables, M-max (Xi, X2,..., Xn) and N- min (X1, X2,... ,Xn)
Problem 1: Let (Xi,..., Xn) denote a random variable from X having a Log-normal density fx (x) = d(L m)/ x, x 〉 0 n(x) - where m is an unknown parameter. Show n-1 Σ'al Ln(X) is a MVU estimator f...
Problem 1: Let (Xi,..., Xn) denote a random variable from X having a Log-normal density fx (x) = d(L m)/ x, x 〉 0 n(x) - where m is an unknown parameter. Show n-1 Σ'al Ln(X) is a MVU estimator for m.
Problem 1: Let (Xi,..., Xn) denote a random variable from X having a Log-normal density fx (x) = d(L m)/ x, x 〉 0 n(x) - where m is an unknown parameter. Show n-1 Σ'al Ln(X) is a...
Problem 4: Consider the problem of estimating the unknown parameter p of a Bernoulli random varia...
Problem 4: Consider the problem of estimating the unknown parameter p of a Bernoulli random variable that describes the probability that a coin toss results in a head. Denote by X the outcome of the jth toss of the coin and let j-1 denote the sample mean. Part I: Use Chebyshev inequality to determine the number of tosses n needed so that P( -pl> 0.01) 0.01 The estimate should be independent of p Part II: Compute ElIX -pl]. Your answer...
Problem 2. Consider a random sample of size n from a two-parameter distribution with parameter 0...
Problem 2. Consider a random sample of size n from a two-parameter distribution with parameter 0 unknown and parameter η known. The population density function is (xi - T) (a) Find the likelihood function simplifying it as much as possible. Likelihood
Problem 3. Consider a random sample X1, X2,..., Xn from a distribution with log-normal pdf (density...
Problem 3. Consider a random sample X1, X2,..., Xn from a distribution with log-normal pdf (density function): for t 0 and 0 otherwise. Both μ and σ 0 are unknown parameters. Find the method of moments estinates μ and σ. Hint: computing moments, change of variable y = Int might be useful.
Problem 3. Consider a random sample X1, X2,..., Xn from a distribution with log-normal pdf (density...
Problem 3. Consider a random sample X1, X2,..., Xn from a distribution with log-normal pdf (density function): for t 0 and 0 otherwise. Both μ and σ 0 are unknown parameters. Find the method of moments estinates μ and σ. Hint: computing moments, change of variable y = Int might be useful.
FF1:18 1H20B B 80 ma2500a16-1 ma2500s14 ma2500a15 ma2500s15 ma2500a17 2. Let Xi, X2 , X10 be...
FF1:18 1H20B B 80 ma2500a16-1 ma2500s14 ma2500a15 ma2500s15 ma2500a17 2. Let Xi, X2 , X10 be a random sample of observations from the N(μ, σ*) distribution where μ is unknown and σ2-10. We reject the null hypothesis Ho : μ-5 in lavour of the alternative hypothesis H1 : μ < 5 if sum of the observations is less than or equal to 35 (a) What is the critical region for the test? (b) Compute the size of the test (c)...
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,...
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,x,Je {0,1)、[i.e., X is a binary sequence of length n] and Y-(Y Rt [i.e. Y is a sequence of n real numbers.) ·X1~" Bernoulli(1/2) ,%) E Ip is the switching probability; when p is small the Markov chain likes to stay in the same state] . conditioned on X, the random variables Yı , . . . , y, are...
Consider a binary response variable y and two explanatory variables xy and x2. The following table contains the parameter estimates of the linear probability model (LPM) and the logit model, wit...
Consider a binary response variable y and two explanatory variables xy and x2. The following table contains the parameter estimates of the linear probability model (LPM) and the logit model, with the associated p-values shown in parentheses. Constant .40 -2.30 x1 x2 0.06 (0.03) 0.36 0.90 (0.03)(0.07) -0.03-0.10 (0.02) (0.01) a. At the 5% significance level, comment on the significance of the variables for both models. Logit gnificant 0 (Not significant x1 x2 b. What is the predicted probability implied...
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
Problem 5 Information
Problem 6: 10 points A) Find moments of first and second order and the variance for variable M in Problem 5 (B) Find expectation and variance for variable N in Problem 5. Problem 5: 10 points Consider n independent variables, {X1, X2,... , Xn} uniformly distributed over the unit interval, (0,1) Introduce two new random variables, M-max (Xi, X2,..., Xn) and N- min (X1, X2,... ,Xn)
Problem 1: Let (Xi,..., Xn) denote a random variable from X having a Log-normal density fx (x) = d(L m)/ x, x 〉 0 n(x) - where m is an unknown parameter. Show n-1 Σ'al Ln(X) is a MVU estimator for m.
Problem 1: Let (Xi,..., Xn) denote a random variable from X having a Log-normal density fx (x) = d(L m)/ x, x 〉 0 n(x) - where m is an unknown parameter. Show n-1 Σ'al Ln(X) is a...
Problem 4: Consider the problem of estimating the unknown parameter p of a Bernoulli random variable that describes the probability that a coin toss results in a head. Denote by X the outcome of the jth toss of the coin and let j-1 denote the sample mean. Part I: Use Chebyshev inequality to determine the number of tosses n needed so that P( -pl> 0.01) 0.01 The estimate should be independent of p Part II: Compute ElIX -pl]. Your answer...
Problem 2. Consider a random sample of size n from a two-parameter distribution with parameter 0 unknown and parameter η known. The population density function is (xi - T) (a) Find the likelihood function simplifying it as much as possible. Likelihood
Problem 3. Consider a random sample X1, X2,..., Xn from a distribution with log-normal pdf (density function): for t 0 and 0 otherwise. Both μ and σ 0 are unknown parameters. Find the method of moments estinates μ and σ. Hint: computing moments, change of variable y = Int might be useful.
Problem 3. Consider a random sample X1, X2,..., Xn from a distribution with log-normal pdf (density function): for t 0 and 0 otherwise. Both μ and σ 0 are unknown parameters. Find the method of moments estinates μ and σ. Hint: computing moments, change of variable y = Int might be useful.
FF1:18 1H20B B 80 ma2500a16-1 ma2500s14 ma2500a15 ma2500s15 ma2500a17 2. Let Xi, X2 , X10 be a random sample of observations from the N(μ, σ*) distribution where μ is unknown and σ2-10. We reject the null hypothesis Ho : μ-5 in lavour of the alternative hypothesis H1 : μ < 5 if sum of the observations is less than or equal to 35 (a) What is the critical region for the test? (b) Compute the size of the test (c)...
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,x,Je {0,1)、[i.e., X is a binary sequence of length n] and Y-(Y Rt [i.e. Y is a sequence of n real numbers.) ·X1~" Bernoulli(1/2) ,%) E Ip is the switching probability; when p is small the Markov chain likes to stay in the same state] . conditioned on X, the random variables Yı , . . . , y, are...
Consider a binary response variable y and two explanatory variables xy and x2. The following table contains the parameter estimates of the linear probability model (LPM) and the logit model, with the associated p-values shown in parentheses. Constant .40 -2.30 x1 x2 0.06 (0.03) 0.36 0.90 (0.03)(0.07) -0.03-0.10 (0.02) (0.01) a. At the 5% significance level, comment on the significance of the variables for both models. Logit gnificant 0 (Not significant x1 x2 b. What is the predicted probability implied...
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