Let (Z, N, +, ·) be an ordered integral domain. Let {x1, x2, . . ....

Let (Z, N, +, ·) be an ordered integral domain. Let {x1, x2, . . . , xn} be a subset of Z. Prove there exists an i, 1 ≤ i ≤ n such that xi ≥ xj for all 1 ≤ j ≤ n. Prove that Z is an infinite set. (Remark: How do you tell if a set is infinite??)

Expert Solution

Colby Messinger answered 2 years ago

Let X1 and X2 be independent standard normal variables X1 ∼ N(0, 1) and X2 ∼...

Let X1 and X2 be independent standard normal variables X1 ∼ N(0, 1) and X2 ∼ N(0, 1). 1) Let Y1 = X12 + X12 and Y2 = X12− X22 . Find the joint p.d.f. of Y1 and Y2, and the marginal p.d.f. of Y1. Are Y1 and Y2 independent? 2) Let W = √X1X2/(X12 +X22) . Find the p.d.f. of W.

Let X1, X2, X3 be independent having N(0,1). Let Y1=(X1-X2)/√2, Y2=(X1+X2-2*X3)/√6, Y3=(X1+X2+X3)/√3. Find the joint pdf...

Let X1, X2, X3 be independent having N(0,1). Let Y1=(X1-X2)/√2, Y2=(X1+X2-2*X3)/√6, Y3=(X1+X2+X3)/√3. Find the joint pdf of Y1, Y2, Y3, and the marginal pdfs.

Let X1 and X2 be uniform on the consecutive integers -n, -(n+1), ... , n-1, n....

Let X1 and X2 be uniform on the consecutive integers -n, -(n+1), ... , n-1, n. Use convolution to find the mass function for X1 + X2.

Let T(x1, x2) = (-x1 + 3x2, x1 - x2) be a transformation. a) Show that...

Let T(x1, x2) = (-x1 + 3x2, x1 - x2) be a transformation. a) Show that T is invertible. b)Find T inverse.

Let the utility function be given by u(x1, x2) = √x1 + x2. Let m be...

Let the utility function be given by u(x1, x2) = √x1 + x2. Let m be the income of the consumer, P1 and P2 the prices of good 1 and good 2, respectively. To simplify, normalize the price of good 1, that is P1 = £1. (a) Write down the budget constraint and illustrate the set of feasible bundles using a figure. (b) Suppose that m = £100 and that P2 = £10. Find the optimal bundle for the consumer....

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2...

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y = (Y1,Y2,Y3)′ using : Multivariate normal distribution properties.

Let X1, X2, . . . , Xn be a random sample of size n from...

Let X1, X2, . . . , Xn be a random sample of size n from a Poisson distribution with unknown mean µ. It is desired to test the following hypotheses H0 : µ = µ0 versus H1 : µ not equal to µ0 where µ0 > 0 is a given constant. Derive the likelihood ratio test statistic

Let X1, X2, · · · , Xn (n ≥ 30) be i.i.d observations from N(µ1,...

Let X1, X2, · · · , Xn (n ≥ 30) be i.i.d observations from N(µ1, σ12 ) and Y1, Y2, · · · , Yn be i.i.d observations from N(µ2, σ22 ). Also assume that X's and Y's are independent. Suppose that µ1, µ2, σ12 , σ22 are unknown. Find an approximate 95% confidence interval for µ1µ2.

Let X1, X2, . . . , Xn iid∼ N (µ, σ2 ). Consider the hypotheses...

Let X1, X2, . . . , Xn iid∼ N (µ, σ2 ). Consider the hypotheses H0 : µ = µ0 and H1 : µ (not equal)= µ0 and the test statistic (X bar − µ0)/ (S/√ n). Note that S has been used as σ is unknown. a. What is the distribution of the test statistic when H0 is true? b. What is the type I error of an α−level test of this type? Prove it. c. What is...

Let X1, X2, X3, . . . be independently random variables such that Xn ∼ Bin(n,...

Let X1, X2, X3, . . . be independently random variables such that Xn ∼ Bin(n, 0.5) for n ≥ 1. Let N ∼ Geo(0.5) and assume it is independent of X1, X2, . . .. Further define T = XN . (a) Find E(T) and argue that T is short proper. (b) Find the pgf of T. (c) Use the pgf of T in (b) to find P(T = n) for n ≥ 0. (d) Use the pgf of...

Question