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a) How many parameters does a mixture of m Gaussians
have?
b) Let x1,...
please work out parts b,c,d with clear steps thanks A mixture of m univariate Gaussians has...
please work out parts b,c,d with clear steps thanks
A mixture of m univariate Gaussians has the PDF: X(x) - where each pi 0 and Σ-i pi-1, and N(x; μ, σ*) = (2πσ2)-1/2 exp (-(x-p?/(2σ2)) exp (-(x-μ)2 a) How many parameters does a mixture of m Gaussians have? b) Let xi, , Vn be n observations drawn from a mixture of m Gaussians. Write down the log-likelihood function. Hint: it should involve two summations c) Let 1 k < m....
No a,b needed. please do c and d with clear steps A mixture of m univariate...
No a,b needed. please do c and d with clear steps
A mixture of m univariate Gaussians has the PDF: X(x) - where each pi 0 and Σ-i pi-1, and N(x; μ, σ*) = (2πσ2)-1/2 exp (-(x-p?/(2σ2)) exp (-(x-μ)2 a) How many parameters does a mixture of m Gaussians have? b) Let xi, , Vn be n observations drawn from a mixture of m Gaussians. Write down the log-likelihood function. Hint: it should involve two summations c) Let 1 k...
2. Suppose that we have n independent observations x1, ,Tn from a normal distribution with mean...
2. Suppose that we have n independent observations x1, ,Tn from a normal distribution with mean μ and variance σ2, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 7-2 log max L(μ, σ*) )-2 log ( max L(u, i) μισ (c) Explain as clearly as you can what happens to T, when our estimate of σ2 is less than 1. (d) Show...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distr...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean μ and variance σ, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 2 lo g max L(μ, σ log | max L( 1) (c) Explain as clearly as you can what happens to T when our estimate of σ2 is less than 1. (d) Show that...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean μ and variance σ, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 2 lo g max L(μ, σ log | max L( 1) (c) Explain as clearly as you can what happens to T when our estimate of σ2 is less than 1. (d) Show that...
01) Let X1, X2, ...., Xm be m independent growth observations on the first type with...
01) Let X1, X2, ...., Xm be m independent growth observations on the first type with E[X:] = u and V(x) = 02. And Let Y, Y2, ...., Yn be n independent growth observations on the second type with E[Y] = u and V(Y) = 402 And let Z1, Z2, ....,Zk be k independent growth observations on the third type with E[Z] = 2u and V(Z) = 302 Show that the estimator û = 32 - 2x - 37 is...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution,...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Wr...
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Write down the expression of the likelihood function We assume that min(x1, . . . ,xm) 〈 n and max(x1, . . . ,xm) 〉 0 5 marks) Find , and give all possible solutions to the equation dL dL -...
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)...
please work out parts b,c,d with clear steps thanks
A mixture of m univariate Gaussians has the PDF: X(x) - where each pi 0 and Σ-i pi-1, and N(x; μ, σ*) = (2πσ2)-1/2 exp (-(x-p?/(2σ2)) exp (-(x-μ)2 a) How many parameters does a mixture of m Gaussians have? b) Let xi, , Vn be n observations drawn from a mixture of m Gaussians. Write down the log-likelihood function. Hint: it should involve two summations c) Let 1 k < m....
No a,b needed. please do c and d with clear steps
A mixture of m univariate Gaussians has the PDF: X(x) - where each pi 0 and Σ-i pi-1, and N(x; μ, σ*) = (2πσ2)-1/2 exp (-(x-p?/(2σ2)) exp (-(x-μ)2 a) How many parameters does a mixture of m Gaussians have? b) Let xi, , Vn be n observations drawn from a mixture of m Gaussians. Write down the log-likelihood function. Hint: it should involve two summations c) Let 1 k...
2. Suppose that we have n independent observations x1, ,Tn from a normal distribution with mean μ and variance σ2, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 7-2 log max L(μ, σ*) )-2 log ( max L(u, i) μισ (c) Explain as clearly as you can what happens to T, when our estimate of σ2 is less than 1. (d) Show...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean μ and variance σ, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 2 lo g max L(μ, σ log | max L( 1) (c) Explain as clearly as you can what happens to T when our estimate of σ2 is less than 1. (d) Show that...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean μ and variance σ, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 2 lo g max L(μ, σ log | max L( 1) (c) Explain as clearly as you can what happens to T when our estimate of σ2 is less than 1. (d) Show that...
01) Let X1, X2, ...., Xm be m independent growth observations on the first type with E[X:] = u and V(x) = 02. And Let Y, Y2, ...., Yn be n independent growth observations on the second type with E[Y] = u and V(Y) = 402 And let Z1, Z2, ....,Zk be k independent growth observations on the third type with E[Z] = 2u and V(Z) = 302 Show that the estimator û = 32 - 2x - 37 is...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Write down the expression of the likelihood function We assume that min(x1, . . . ,xm) 〈 n and max(x1, . . . ,xm) 〉 0 5 marks) Find , and give all possible solutions to the equation dL dL -...
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)...
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