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,z >0 Where θ > 0, Suppose there are two independent variables Xi; 2 belonging to this distri- 1,...
Suppose X1, X2,... are independent Geometric (number of trials) random variables where Xi ~ Geome...
Suppose X1, X2,... are independent Geometric (number of trials)
random variables where Xi ~ Geometric(p = 1/i^2)
a) It is easily shown that Xn converges to a for some constant
a. Name it.
b) According to the Borel-Cantelli Lemmas, does Xn almost surely
converge to a?
Suppose Xi, X2, are independent Geometric (number of trials) random variables where x,~ Geometric(pal+) |. a) It is easily shown that Xa for some constant a. Name it. b) According to the Borel-Cantelli Lemmas,...
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 :...
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 : θ 1 . We have two tests: where 0<c<1 (a) Show that the power functions of the two tests are A(0)-1-(0.9)θ and β2(0)-1 + d|θ Inc-1), respectively. (b) Calculate the size of the φι test. Then, find the value of c that gives the same size for the φ2 test. (c) Is фг a most powerful test of...
Suppose that Xi, X2,..., Xn are independent random variables (not iid) with densities x, (x^, where...
Suppose that Xi, X2,..., Xn are independent random variables (not iid) with densities x, (x^, where 6, > 0, for i-1, 2, , n. versus H1: not Ho (c) Suppose Ho is true so that the common distribution of X1, X2,..., Xn, now viewed as being conditional on 6, is described by where θ > 0. Identify a conjugate prior for 0. Specify any hyperparameters in your prior (pick values for fun if you want). Show how to carry out...
1. Let Xi, X2,... be independent random variables each with the standard normal distribution, and for...
1. Let Xi, X2,... be independent random variables each with the standard normal distribution, and for each n 2 0 let Sn-1 Xi. Use importance sampling to obtain good estimates for each of the following probabilities: (a) Pfmaxn<100 Sn> 10; and (b) Pímaxns100 Sn > 30) HINTS: The basic identity of importance sampling implies that d.P n100 where Po is the probability measure under which the random variables Xi, X2,... are independent normals with mean 0 amd variance 1. The...
Suppose Xi, X2,.. are independent Geometric (number of trials) random variables where ~Geometric ...
Suppose Xi, X2,.. are independent Geometric (number of trials) random variables where ~Geometric p 1- a) It is easily shown that Xfor some constant a. Name it. a= b) According to the Borel-Cantelli Lemmas, does a.s /n In other words, will there eventually reach a point in the sequence of random variables where every X a?
Suppose Xi, X2,.. are independent Geometric (number of trials) random variables where ~Geometric p 1- a) It is easily shown that Xfor some constant...
Suppose Z^ - hZ1 A where Z1,A are independent random variables with mean 0 Part a)...
Suppose Z^ - hZ1 A where Z1,A are independent random variables with mean 0 Part a) If σ-4, σ:1-1 and φ- 2? 0.3 , what is σ Part b) If σ 6 and φ-0.7, what is ơỈ in order that σ}i-Ž ?
You may use the following facts to answer the questions below Fact 1: Suppose that Xi. . . . , X, are independent and X.* GAM (θ.k.) for -1 -1 Fact 2: If Y GAM(0,n aYGAM(ab,n) for any number a &g...
You may use the following facts to answer the questions below Fact 1: Suppose that Xi. . . . , X, are independent and X.* GAM (θ.k.) for -1 -1 Fact 2: If Y GAM(0,n aYGAM(ab,n) for any number a >0 1. Suppose that V-GAM(1m) and let lPa θν, where θ > 0. (a) Show that, for any given positive number a, P> a) is an increasing function of (b) What is the probability distribution of W? (c) Would you...
2. Suppose Xi ~ N(8,02) where θ > 0. (a) Show that s--(x, Σ¡! xi) is...
2. Suppose Xi ~ N(8,02) where θ > 0. (a) Show that s--(x, Σ¡! xi) is a sufficient statistic of θ where X is the sample mean. (b) Is S minimal sufficient? (c) Can you find a non-constant function g(.) such that g(S) is an ancillary statistic?
et Xi and X2 be independent random variables with common distribution NORM(8,1), where θ is a...
et Xi and X2 be independent random variables with common distribution NORM(8,1), where θ is a real number. Let Y = X1 + X a) Find the p.d.f. of Y. 4. L 2. b) Use the appropriate table to find P(Y> 20 + 2)
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho...
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 : θ 1 . We have two tests: where 0<c<1 (a) Show that the power functions of the two tests are A(0)-1-(0.9)θ and β2(0)-1 + d|θ Inc-1), respectively. (b) Calculate the size of the φι test. Then, find the value of c that gives the same size for the φ2 test. (c) Is фг a most powerful test of...
Suppose X1, X2,... are independent Geometric (number of trials)
random variables where Xi ~ Geometric(p = 1/i^2)
a) It is easily shown that Xn converges to a for some constant
a. Name it.
b) According to the Borel-Cantelli Lemmas, does Xn almost surely
converge to a?
Suppose Xi, X2, are independent Geometric (number of trials) random variables where x,~ Geometric(pal+) |. a) It is easily shown that Xa for some constant a. Name it. b) According to the Borel-Cantelli Lemmas,...
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 : θ 1 . We have two tests: where 0<c<1 (a) Show that the power functions of the two tests are A(0)-1-(0.9)θ and β2(0)-1 + d|θ Inc-1), respectively. (b) Calculate the size of the φι test. Then, find the value of c that gives the same size for the φ2 test. (c) Is фг a most powerful test of...
Suppose that Xi, X2,..., Xn are independent random variables (not iid) with densities x, (x^, where 6, > 0, for i-1, 2, , n. versus H1: not Ho (c) Suppose Ho is true so that the common distribution of X1, X2,..., Xn, now viewed as being conditional on 6, is described by where θ > 0. Identify a conjugate prior for 0. Specify any hyperparameters in your prior (pick values for fun if you want). Show how to carry out...
1. Let Xi, X2,... be independent random variables each with the standard normal distribution, and for each n 2 0 let Sn-1 Xi. Use importance sampling to obtain good estimates for each of the following probabilities: (a) Pfmaxn<100 Sn> 10; and (b) Pímaxns100 Sn > 30) HINTS: The basic identity of importance sampling implies that d.P n100 where Po is the probability measure under which the random variables Xi, X2,... are independent normals with mean 0 amd variance 1. The...
Suppose Xi, X2,.. are independent Geometric (number of trials) random variables where ~Geometric p 1- a) It is easily shown that Xfor some constant a. Name it. a= b) According to the Borel-Cantelli Lemmas, does a.s /n In other words, will there eventually reach a point in the sequence of random variables where every X a?
Suppose Xi, X2,.. are independent Geometric (number of trials) random variables where ~Geometric p 1- a) It is easily shown that Xfor some constant...
Suppose Z^ - hZ1 A where Z1,A are independent random variables with mean 0 Part a) If σ-4, σ:1-1 and φ- 2? 0.3 , what is σ Part b) If σ 6 and φ-0.7, what is ơỈ in order that σ}i-Ž ?
You may use the following facts to answer the questions below Fact 1: Suppose that Xi. . . . , X, are independent and X.* GAM (θ.k.) for -1 -1 Fact 2: If Y GAM(0,n aYGAM(ab,n) for any number a >0 1. Suppose that V-GAM(1m) and let lPa θν, where θ > 0. (a) Show that, for any given positive number a, P> a) is an increasing function of (b) What is the probability distribution of W? (c) Would you...
2. Suppose Xi ~ N(8,02) where θ > 0. (a) Show that s--(x, Σ¡! xi) is a sufficient statistic of θ where X is the sample mean. (b) Is S minimal sufficient? (c) Can you find a non-constant function g(.) such that g(S) is an ancillary statistic?
et Xi and X2 be independent random variables with common distribution NORM(8,1), where θ is a real number. Let Y = X1 + X a) Find the p.d.f. of Y. 4. L 2. b) Use the appropriate table to find P(Y> 20 + 2)
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 : θ 1 . We have two tests: where 0<c<1 (a) Show that the power functions of the two tests are A(0)-1-(0.9)θ and β2(0)-1 + d|θ Inc-1), respectively. (b) Calculate the size of the φι test. Then, find the value of c that gives the same size for the φ2 test. (c) Is фг a most powerful test of...
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