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3. Suppose you have X Binom(n, p) where n is known and p is unknown. Typically,...
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows...
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n (b) [1 point] If K is a biased estimator for o?, state the function of K that would make it an unbiased...
2. Suppose you decide to randomly generate numbers from X ~ Unif (0,0). Your friend will...
2. Suppose you decide to randomly generate numbers from X ~ Unif (0,0). Your friend will ask for n numbers and then use this information to guess what value you (secretly) chose for θ. Typically, one might use θMLE-max Xi-X, to estimate θ. Your friend, however, has meganumerophobia, and is afraid to say the maximum number in the random sample. Instead he'll say the second largest number: θ-Xn-1. Determine the bias of this estimator by carefully finding the density function...
2. Suppose you decide to randomly generate numbers from X ~ Unif(0, ). Your friend will...
2. Suppose you decide to randomly generate numbers from X ~ Unif(0, ). Your friend will ask for n numbers and then use this information to guess what value you (secretly) chose for θ. Typically, one might use alLE = max Xi = X, to estimate θ. Your friend, however, has meganumerophobia, and is afraid to say the maximum number in the random sample. Instead, he'll say the second largest number: θ = Xn-1. Determine the bias of this estimator...
3. Let X~ Bin(n,p) with n known (a) State the parameter space for the mode b)...
3. Let X~ Bin(n,p) with n known (a) State the parameter space for the mode b) State EX] and V[x]. (c) Is p an unbiased estimator for the population proportion p? Show why or why not (d) To estimate the variance of X, we generally use θ 2Pl1 ow is a estimator for V지. (e) Modify 0 from part (b) to form an unbiased estimator for V[X ].
Let X1, ..., X., be i.i.d random variables N(u, 02) where u is known parameter and...
Let X1, ..., X., be i.i.d random variables N(u, 02) where u is known parameter and o2 is the unknown parameter. Let y() = 02. (i) Find the CRLB for yo?). (ii) Recall that S2 is an unbiased estimator for o2. Compare the Var(S2) to that of the CRLB for
You observe sarriples X1....,x. Ber(@) where 0 € (0,1) is an unknown parameter. Suppose that is...
You observe sarriples X1....,x. Ber(@) where 0 € (0,1) is an unknown parameter. Suppose that is much larger than 1 so we have access to many samples from the specified distribution. Consider three candidate estimators for 0. . X.-1x . 0.5 In the next three questions, you will consider potential strengths and weaknesses of these estimators. In this particular section (just for now), the phrase "efficiently computable" refers to the existence of an explicit formula. More precisely, here we say...
2). I. In this problem, we explore more estimators for N4ơ a. Typically, people use /1...
2). I. In this problem, we explore more estimators for N4ơ a. Typically, people use /1 X as an estimator for μ. You might also use μ2 or is -2x1 + 12 2 (x1+2X2 +3xs+...+nXn), Show that all three of these estimato b. In class, we showed that σ-T ( i-1 x,-X) is a biased estimator for ơ2 when both and Tt σ are unknown. Suppose, however, that μ is known and so we can use σ Show that σ2...
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...
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 -...
Suppose you have a random sample {X1, X2, X3} of size n = 3. Consider the following three possible estimators for the p...
Suppose you have a random sample {X1, X2, X3} of size n = 3. Consider the following three possible estimators for the population mean u and variance o2 Дi 3D (X1+ X2+ X3)/3 Ti2X1/4 X2/2 X3/4 Дз — (Х+ X,+ X3)/4 (a) What is the bias associated with each estimator? (b) What is the variance associated with each estimator? (c) Does the fact that Var(i3) < Var(1) contradict the statement that X is the minimum variance unbiased estimator? Why or...
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n (b) [1 point] If K is a biased estimator for o?, state the function of K that would make it an unbiased...
2. Suppose you decide to randomly generate numbers from X ~ Unif (0,0). Your friend will ask for n numbers and then use this information to guess what value you (secretly) chose for θ. Typically, one might use θMLE-max Xi-X, to estimate θ. Your friend, however, has meganumerophobia, and is afraid to say the maximum number in the random sample. Instead he'll say the second largest number: θ-Xn-1. Determine the bias of this estimator by carefully finding the density function...
2. Suppose you decide to randomly generate numbers from X ~ Unif(0, ). Your friend will ask for n numbers and then use this information to guess what value you (secretly) chose for θ. Typically, one might use alLE = max Xi = X, to estimate θ. Your friend, however, has meganumerophobia, and is afraid to say the maximum number in the random sample. Instead, he'll say the second largest number: θ = Xn-1. Determine the bias of this estimator...
3. Let X~ Bin(n,p) with n known (a) State the parameter space for the mode b) State EX] and V[x]. (c) Is p an unbiased estimator for the population proportion p? Show why or why not (d) To estimate the variance of X, we generally use θ 2Pl1 ow is a estimator for V지. (e) Modify 0 from part (b) to form an unbiased estimator for V[X ].
Let X1, ..., X., be i.i.d random variables N(u, 02) where u is known parameter and o2 is the unknown parameter. Let y() = 02. (i) Find the CRLB for yo?). (ii) Recall that S2 is an unbiased estimator for o2. Compare the Var(S2) to that of the CRLB for
You observe sarriples X1....,x. Ber(@) where 0 € (0,1) is an unknown parameter. Suppose that is much larger than 1 so we have access to many samples from the specified distribution. Consider three candidate estimators for 0. . X.-1x . 0.5 In the next three questions, you will consider potential strengths and weaknesses of these estimators. In this particular section (just for now), the phrase "efficiently computable" refers to the existence of an explicit formula. More precisely, here we say...
2). I. In this problem, we explore more estimators for N4ơ a. Typically, people use /1 X as an estimator for μ. You might also use μ2 or is -2x1 + 12 2 (x1+2X2 +3xs+...+nXn), Show that all three of these estimato b. In class, we showed that σ-T ( i-1 x,-X) is a biased estimator for ơ2 when both and Tt σ are unknown. Suppose, however, that μ is known and so we can use σ Show that σ2...
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...
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 -...
Suppose you have a random sample {X1, X2, X3} of size n = 3. Consider the following three possible estimators for the population mean u and variance o2 Дi 3D (X1+ X2+ X3)/3 Ti2X1/4 X2/2 X3/4 Дз — (Х+ X,+ X3)/4 (a) What is the bias associated with each estimator? (b) What is the variance associated with each estimator? (c) Does the fact that Var(i3) < Var(1) contradict the statement that X is the minimum variance unbiased estimator? Why or...
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