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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...
Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent...
Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent ran- dom variables, we can find joint probabilities by multiplying together th ndividual probabilities. For example This should remind you the discussion on statistical independence of random variables that can be found in the course book (see page 22) Answer the following questions a...
3. Suppose Xi, X2, and X are independent random variables drawn from a binomial distribution with parameters p and n. The observed values are Xi -3, X2-4, and (a) Suppose n 12 and p is unknown. What...
3. Suppose Xi, X2, and X are independent random variables drawn from a binomial distribution with parameters p and n. The observed values are Xi -3, X2-4, and (a) Suppose n 12 and p is unknown. What is the maximum likelihood estimator (b) Suppose p - 0.4 and n is unknown. What is the maximum likelihood estimator for p? for n? (Note: Since n is discrete you can't use calculus for this; just write the formula and use trial and...
Let X1 and X2 be independent n(0,1) random variables. Find the pdf of (X1 - X2)^2/2
Let X1 and X2 be independent n(0,1) random variables. Find the pdf of (X1 - X2)^2/2
Let X1 and X2 be independent random variables so X1~ N(u,1) and X2 N(u,4) Where u...
Let X1 and X2 be independent random variables so X1~ N(u,1) and X2 N(u,4) Where u R a) Show that the likelihood for , given that X1 = x1 and X2 = xz is 8 4T b) Show, that the maxium likelihood estimate for u is 4x1+ x2 и (х, х2) e) Show that СтN -("x"x) .я d) and enter a formula for the 95% confidence interval for
Let X1 and X2 be independent random variables so X1~ N(u,1) and...
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that...
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .
Problem 5: 10 points Consider n independent variables, {X1, X2,... , Xn) uniformly distributed over the...
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 (X1, X2,..., Xn) and N -min (X1, X2,..., Xn) 1. Find the joint distribution of a pair (M,N) 2. Derive the CDF and density for M 3. Derive the CDF and density for N.
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)...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
Let X1,X2,...,Xn be iid exponential random variables with unknown mean β. (b) Find the maximum likelihood...
Let X1,X2,...,Xn be iid exponential random variables with unknown mean β. (b) Find the maximum likelihood estimator of β. (c) Determine whether the maximum likelihood estimator is unbiased for β. (d) Find the mean squared error of the maximum likelihood estimator of β. (e) Find the Cramer-Rao lower bound for the variances of unbiased estimators of β. (f) What is the UMVUE (uniformly minimum variance unbiased estimator) of β? What is your reason? (g) Determine the asymptotic distribution of the...
Suppose X1,X2,…,Xn represent the outcomes of n independent Bernoulli trials, each with success probability p. Note...
Suppose X1,X2,…,Xn represent the outcomes of n independent
Bernoulli trials, each with success probability p. Note that we can
write the Bernoulli distribution as:
Suppose X1 2 X, represent the outcomes of n independent Bernou i als, each with success probabil ,p. Note that we can writ e the Bernoulǐ distribution as 0,1 otherwise Given the Bernoulli distributional family and the iid sample of X,'s, the likelihood function is: -1 a. Find an expression for p, the MLE of p...
Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent ran- dom variables, we can find joint probabilities by multiplying together th ndividual probabilities. For example This should remind you the discussion on statistical independence of random variables that can be found in the course book (see page 22) Answer the following questions a...
3. Suppose Xi, X2, and X are independent random variables drawn from a binomial distribution with parameters p and n. The observed values are Xi -3, X2-4, and (a) Suppose n 12 and p is unknown. What is the maximum likelihood estimator (b) Suppose p - 0.4 and n is unknown. What is the maximum likelihood estimator for p? for n? (Note: Since n is discrete you can't use calculus for this; just write the formula and use trial and...
Let X1 and X2 be independent random variables so X1~ N(u,1) and X2 N(u,4) Where u R a) Show that the likelihood for , given that X1 = x1 and X2 = xz is 8 4T b) Show, that the maxium likelihood estimate for u is 4x1+ x2 и (х, х2) e) Show that СтN -("x"x) .я d) and enter a formula for the 95% confidence interval for
Let X1 and X2 be independent random variables so X1~ N(u,1) and...
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .
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 (X1, X2,..., Xn) and N -min (X1, X2,..., Xn) 1. Find the joint distribution of a pair (M,N) 2. Derive the CDF and density for M 3. Derive the CDF and density for N.
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)...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
Suppose X1,X2,…,Xn represent the outcomes of n independent
Bernoulli trials, each with success probability p. Note that we can
write the Bernoulli distribution as:
Suppose X1 2 X, represent the outcomes of n independent Bernou i als, each with success probabil ,p. Note that we can writ e the Bernoulǐ distribution as 0,1 otherwise Given the Bernoulli distributional family and the iid sample of X,'s, the likelihood function is: -1 a. Find an expression for p, the MLE of p...
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