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1 Bookmark this page Setup: For all problems on this page, suppose you have data X],...,x...
Bookmark this page Setup: All problems on this page will follow the definitions here: Let X,...
Bookmark this page Setup: All problems on this page will follow the definitions here: Let X, Y be two Bernoulli random variables and let P a r = = = P(X = 1) (the probability that X = 1) P(Y = 1) (the probability that Y = 1) P(X = 1, Y = 1) (the probability that both X = 1 and Y = 1). Let (X1,Y1), ... ,(Xn, Yn) be a sample of n i.i.d. copies of (X, Y)....
5. A confidence interval for Poisson variables Bookmark this page (a) 2 points possible (graded) Let...
5. A confidence interval for Poisson variables Bookmark this page (a) 2 points possible (graded) Let X1,..., Xn bei.i.d. Poisson random variables with parameter 1 > 0 and denote by Xn their empirical average, Xn=1 İx; and (bn), such that an (Xn-bn) converges in distribution to a standard Gaussian random variable Find two sequences (an) Z~ N(0,1). an =
Problem 4 True or False A Bookmark this page Instructions: Be very careful with the multiple choice questions below. Some are "choose all that apply," and many tests your knowledge of when pa...
Problem 4 True or False A Bookmark this page Instructions: Be very careful with the multiple choice questions below. Some are "choose all that apply," and many tests your knowledge of when particular statements apply As in the rest of this exam, only your last submission will count. 1 point possible (graded, results hidden) The likelihood ratio test is used to obtain a test with non-asymptotic level o True O False Submit You have used 0 of 3 attempts Save...
2. Biased and unbiased estimation for variance of Bernoulli variables A Bookmark this page 2 points...
2. Biased and unbiased estimation for variance of Bernoulli variables A Bookmark this page 2 points possible (graded) Let X1, X, bed. Bernoull random variables, with unknown parameter PE (0,1). The aim of this exercise is to estimate the common variance of the X First, recall what Var (X) is for Bernoulli random variables. Var (X) - Let X, be the sample average of the Xi. X. - 3x Interested in finding an estimator for Var(X), and propose to use...
4. Setup: Suppose you have observations X1,X2,X3,X4,X5 which are i.i.d. draws from a Gaussian distribution with...
4.
Setup:
Suppose you have observations X1,X2,X3,X4,X5 which are i.i.d.
draws from a Gaussian distribution with unknown mean μ and unknown
variance σ2.
Given Facts:
You are given the following:
15∑i=15Xi=0.90,15∑i=15X2i=1.31
Bookmark this page Setup: Suppose you have observations X1, X2, X3, X4, X5 which are i.i.d. draws from a Gaussian distribution with unknown mean u and unknown variance o? Given Facts: You are given the following: x=030, =1:1 Choose a test 1 point possible (graded, results hidden) To test...
11. Chi-Squared Test for a Family of Discrete Distributions A Bookmark this page In the problems...
11. Chi-Squared Test for a Family of Discrete Distributions A Bookmark this page In the problems on this page you will apply the goodness of fit test to determine whether or not a sample has a binomial distribution So far, we have used the x test to determine if our data had a categorical distribution with specific parameters (e.s uniform on an set). element For the problems on this page, we extend the discussion on x tests beyond what was...
3. Method of moments estimators Bookmark this page For each of the following distributions, give the...
3. Method of moments estimators Bookmark this page For each of the following distributions, give the method of moments estimator in terms of the sample averages Xn and X, assuming we have access to n i.i.d. observations X1,...,xn. In other words, express the parameters as functions of E [X1 and E[X] and then apply these functions to Xn and X (b) 1 point possible (graded) X; Poiss (), > 0, which means that each X1 has the pmf PX(X =...
Square of a standard normal: let X1, ..., Xn ~ X be i.i.d. standard normal variables....
Square of a standard normal: let X1, ..., Xn ~ X be i.i.d. standard normal variables. What is the mean E[X2] and variance Var [X2] of the random variable x?? E[X2] = Var [X2]
Assume that you have random variable X with pdf or pmf f(x; θ1, . . ....
Assume that you have random variable X with pdf or pmf f(x; θ1, . . . , θk). Let X1, . . . , Xn be a random sample from X. Then Mj = (1/n)Xn i=1 (Xi)j is known as the j-th sample moment of the sample. The moment estimators of θ1, . . . , θk, denoted by ˜θ1, . . . , ˜θk, are the values of θ1, . . . , θk which solve the k equations...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y)...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
Bookmark this page Setup: All problems on this page will follow the definitions here: Let X, Y be two Bernoulli random variables and let P a r = = = P(X = 1) (the probability that X = 1) P(Y = 1) (the probability that Y = 1) P(X = 1, Y = 1) (the probability that both X = 1 and Y = 1). Let (X1,Y1), ... ,(Xn, Yn) be a sample of n i.i.d. copies of (X, Y)....
5. A confidence interval for Poisson variables Bookmark this page (a) 2 points possible (graded) Let X1,..., Xn bei.i.d. Poisson random variables with parameter 1 > 0 and denote by Xn their empirical average, Xn=1 İx; and (bn), such that an (Xn-bn) converges in distribution to a standard Gaussian random variable Find two sequences (an) Z~ N(0,1). an =
Problem 4 True or False A Bookmark this page Instructions: Be very careful with the multiple choice questions below. Some are "choose all that apply," and many tests your knowledge of when particular statements apply As in the rest of this exam, only your last submission will count. 1 point possible (graded, results hidden) The likelihood ratio test is used to obtain a test with non-asymptotic level o True O False Submit You have used 0 of 3 attempts Save...
2. Biased and unbiased estimation for variance of Bernoulli variables A Bookmark this page 2 points possible (graded) Let X1, X, bed. Bernoull random variables, with unknown parameter PE (0,1). The aim of this exercise is to estimate the common variance of the X First, recall what Var (X) is for Bernoulli random variables. Var (X) - Let X, be the sample average of the Xi. X. - 3x Interested in finding an estimator for Var(X), and propose to use...
4.
Setup:
Suppose you have observations X1,X2,X3,X4,X5 which are i.i.d.
draws from a Gaussian distribution with unknown mean μ and unknown
variance σ2.
Given Facts:
You are given the following:
15∑i=15Xi=0.90,15∑i=15X2i=1.31
Bookmark this page Setup: Suppose you have observations X1, X2, X3, X4, X5 which are i.i.d. draws from a Gaussian distribution with unknown mean u and unknown variance o? Given Facts: You are given the following: x=030, =1:1 Choose a test 1 point possible (graded, results hidden) To test...
11. Chi-Squared Test for a Family of Discrete Distributions A Bookmark this page In the problems on this page you will apply the goodness of fit test to determine whether or not a sample has a binomial distribution So far, we have used the x test to determine if our data had a categorical distribution with specific parameters (e.s uniform on an set). element For the problems on this page, we extend the discussion on x tests beyond what was...
3. Method of moments estimators Bookmark this page For each of the following distributions, give the method of moments estimator in terms of the sample averages Xn and X, assuming we have access to n i.i.d. observations X1,...,xn. In other words, express the parameters as functions of E [X1 and E[X] and then apply these functions to Xn and X (b) 1 point possible (graded) X; Poiss (), > 0, which means that each X1 has the pmf PX(X =...
Square of a standard normal: let X1, ..., Xn ~ X be i.i.d. standard normal variables. What is the mean E[X2] and variance Var [X2] of the random variable x?? E[X2] = Var [X2]
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
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