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Additional Problem 9. Suppose ZN(0, 1). (a) Show that the mgf of Z is M2(t)-er. Hint:...
Problem 2 [17 points]. Transformations! a) (5 points) Suppose the time, W, it takes to complete...
Problem 2 [17 points]. Transformations! a) (5 points) Suppose the time, W, it takes to complete a technical task at a workshop has probability density function -w/2 f(w)y 0, 0, otherwise Using the appropriate transformation methods, find the density function for the a time it takes two workers to complete this technical task: S Wi + Ws b) (5 points) Derive the moment generating function of a standard normal randon variable. Use point form to explain each step in your...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is uni...
Number Theory
13 and 14 please!
13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
/20 marks totall Problem 3 Suppose that a rv Y has mgf m(t)- (a) (1- bt)'...
/20 marks totall Problem 3 Suppose that a rv Y has mgf m(t)- (a) (1- bt)' Differentiate this mgf twice and thereby obtain the mean and variance of Y. 5 marks/ Suppose m(1) is the mgf of a rv W. Let r(t) be the natural logarithm of mo. i.e. r) logm(t). Find rand "(t), and express (0) and "(0) in terms of EW and VarW. (b) 5 marks/ (c) Use the result in (b) to find the mean and variance...
Problem 1 Let Xi, ,Xn be a random sample from a Normal distribution with mean μ and variance 1.e Answer the following questions for 8 points total (a) Derive the moment generating function of the dis...
Problem 1 Let Xi, ,Xn be a random sample from a Normal distribution with mean μ and variance 1.e Answer the following questions for 8 points total (a) Derive the moment generating function of the distribution. (1 point). Hint: use the fact that PDF of a density always integrates to 1. (b) Show that the mean of the distribution is u (proof needed). (1 point) (c) Using random sample X1, ,Xn to derive the maximum likelihood estimator of μ (2...
The moment generating function (MGF) for a certain probability distribution is given by 2 (2 + 2) , M(t) = R. t 2 Suppose Xi, X2, are iid random variables with this distribution. Let Sn -Xi+ (a) Show...
The moment generating function (MGF) for a certain probability distribution is given by 2 (2 + 2) , M(t) = R. t 2 Suppose Xi, X2, are iid random variables with this distribution. Let Sn -Xi+ (a) Show that Var(X) =3/2, i = 1,2. (b) Give the MGF of Sn/v3n/2. (c) Evaluate the limit of the MGF in (b) for n → 0.
The moment generating function (MGF) for a certain probability distribution is given by 2 (2 + 2)...
Problem 2. Show that S = {(x, y, z) ER + = 1) is a smooth...
Problem 2. Show that S = {(x, y, z) ER + = 1) is a smooth surface. (hint: use parametrization (u) = (COS11, , sin u). On what domain??)
Only 1-6) N(4,) "x.xx be a random sample from variance, respectively. In order to show that...
Only 1-6)
N(4,) "x.xx be a random sample from variance, respectively. In order to show that and let X and S be sample mean and sample 1. Let and 5 are independent, tollow the steps below. 1-1) Use the change of variable technique =nx-x,- x and show the joint pdf of ,X,,X is (n-1) n- exp f(,x) 20 2a av2 Use Jacobian for n x n variable transformation 1-2) Use the fact that N(u,a n), and show that the conditional...
Problem 2. Problem 1 doesn't need to be done, it's here for reference 166 Branching processes...
Problem 2. Problem 1 doesn't need to be done, it's here for
reference
166 Branching processes is a branching process whose .... is a branchin the result zes have mean μ (s l ) and variance σ 2, then var( ZnJun of Problem 9.6.1 to show that, if Zo. z 2. Use ditioning on the value of Zm, show th ose fa outition theorem and conditioning on the value of Z 9.6 Problems I. Let X1 , X2. . ....
1. Let a,b ER with a < b. In this problem we are going to prove...
1. Let a,b ER with a < b. In this problem we are going to prove that the open interval (a, b) containes infinitely many rational numbers by following these steps (a) First let NEN be an arbitrary rational number. Use the density of the rational numbers to show that (a, b) contains N rational numbers. There is a hint about this in the lecture on the density of rationals.) (b) Now uppose that there are finitely many rational numbers...
Problem 2 [17 points]. Transformations! a) (5 points) Suppose the time, W, it takes to complete a technical task at a workshop has probability density function -w/2 f(w)y 0, 0, otherwise Using the appropriate transformation methods, find the density function for the a time it takes two workers to complete this technical task: S Wi + Ws b) (5 points) Derive the moment generating function of a standard normal randon variable. Use point form to explain each step in your...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
Number Theory
13 and 14 please!
13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
/20 marks totall Problem 3 Suppose that a rv Y has mgf m(t)- (a) (1- bt)' Differentiate this mgf twice and thereby obtain the mean and variance of Y. 5 marks/ Suppose m(1) is the mgf of a rv W. Let r(t) be the natural logarithm of mo. i.e. r) logm(t). Find rand "(t), and express (0) and "(0) in terms of EW and VarW. (b) 5 marks/ (c) Use the result in (b) to find the mean and variance...
Problem 1 Let Xi, ,Xn be a random sample from a Normal distribution with mean μ and variance 1.e Answer the following questions for 8 points total (a) Derive the moment generating function of the distribution. (1 point). Hint: use the fact that PDF of a density always integrates to 1. (b) Show that the mean of the distribution is u (proof needed). (1 point) (c) Using random sample X1, ,Xn to derive the maximum likelihood estimator of μ (2...
The moment generating function (MGF) for a certain probability distribution is given by 2 (2 + 2) , M(t) = R. t 2 Suppose Xi, X2, are iid random variables with this distribution. Let Sn -Xi+ (a) Show that Var(X) =3/2, i = 1,2. (b) Give the MGF of Sn/v3n/2. (c) Evaluate the limit of the MGF in (b) for n → 0.
The moment generating function (MGF) for a certain probability distribution is given by 2 (2 + 2)...
Problem 2. Show that S = {(x, y, z) ER + = 1) is a smooth surface. (hint: use parametrization (u) = (COS11, , sin u). On what domain??)
Only 1-6)
N(4,) "x.xx be a random sample from variance, respectively. In order to show that and let X and S be sample mean and sample 1. Let and 5 are independent, tollow the steps below. 1-1) Use the change of variable technique =nx-x,- x and show the joint pdf of ,X,,X is (n-1) n- exp f(,x) 20 2a av2 Use Jacobian for n x n variable transformation 1-2) Use the fact that N(u,a n), and show that the conditional...
Problem 2. Problem 1 doesn't need to be done, it's here for
reference
166 Branching processes is a branching process whose .... is a branchin the result zes have mean μ (s l ) and variance σ 2, then var( ZnJun of Problem 9.6.1 to show that, if Zo. z 2. Use ditioning on the value of Zm, show th ose fa outition theorem and conditioning on the value of Z 9.6 Problems I. Let X1 , X2. . ....
1. Let a,b ER with a < b. In this problem we are going to prove that the open interval (a, b) containes infinitely many rational numbers by following these steps (a) First let NEN be an arbitrary rational number. Use the density of the rational numbers to show that (a, b) contains N rational numbers. There is a hint about this in the lecture on the density of rationals.) (b) Now uppose that there are finitely many rational numbers...
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