1. Let α < β be real numbers and N ∈ N. (a). Show that if...

1. Let α < β be real numbers and N ∈ N.

(a). Show that if β − α > N then there are at least N distinct integers strictly between β and α.
(b). Show that if β > α are real numbers then there is a rational number q ∈ Q such β > q > α.

***********************************************************************************************

2. Let x, y, z be real numbers.The absolute value of x is defined by

|x|= x, if x ≥ 0 ,

−x, if x < 0

Define the following statements. Assume that the product of two positive real numbers is positive.

(a). Show that |x−y| = |y−x|.

(b). Show that if z<0 and x ≤ y, then zy ≤ zx.

(c). Show that |x+y| ≤ |x|+|y|.

***********************************************************************************************

Answer both questions plz. Will rate.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α...

Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α ≠ Q 2) if r ∈ α and s ∈ Q satisfies s < r, then s ∈ α. 3) if r ∈ α, then there exists s ∈ Q with s > r and s ∈ α. Let α + β = {r + s | r ∈ α and s ∈ β}. Show that the set of all cuts R with the...

Using Baynesian estimation. 1. Let X is Poi(Ꝋ). Let Ꝋ be Γ(α, β). Show that the...

Using Baynesian estimation. 1. Let X is Poi(Ꝋ). Let Ꝋ be Γ(α, β). Show that the marginal pmf of X (the compound distribution) is k1(x) = (Γ (α + x) β^x) / (Γ(α) x! (1 + β)^(α+x ); x = 0, 1, 2, 3, …, which is a generalization of the negative binomial distribution.

A = [(α −β −β), (−β α −β), (−β −β α) ] (each row is in...

A = [(α −β −β), (−β α −β), (−β −β α) ] (each row is in paranthesis) (c) Find the algebraic multiplicity and the geometric multiplicity of every eigenvalue of A. (d) Justify if the matrix A is diagonalizable.

1. Homework 4 Due: 2/28 Let X1,...,Xn i.i.d. Gamma(α,β) with α > 0, β > 0...

1. Homework 4 Due: 2/28 Let X1,...,Xn i.i.d. Gamma(α,β) with α > 0, β > 0 (a) Assume both α and β are unknown, find their momthod of moment estimators: αˆMOM and βˆMOM. (b) Assume α is known and β is unknown, find the maximum likelihood estimation for β. 2.

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0....

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0. (b) Show that A is symmetric if and only if A^2= AA^T

Let A = {a1,...,an} be a set of real numbers such that ai >= 1 for...

Let A = {a1,...,an} be a set of real numbers such that ai >= 1 for all i, and let I be an open interval of length 1. Use Sperner’s Theorem to prove an upper bound for the number of subsets of A whose elements sum to a number inside the interval I.

Let α be a root of x^2 +1 ∈ Z3[x]. Show that Z3(α) = {a+bα :...

Let α be a root of x^2 +1 ∈ Z3[x]. Show that Z3(α) = {a+bα : a, b ∈ Z3} is isomorphic to Z3[x]/(x^2 + 1).

Let k ≥ 2. Use that R (the real numbers) is complete to show R^k is...

Let k ≥ 2. Use that R (the real numbers) is complete to show R^k is complete.

1. Let W1, . . . , Wn be i.i.d. N(µ, σ2). A 100(1 − α)%...

1. Let W1, . . . , Wn be i.i.d. N(µ, σ2). A 100(1 − α)% confidence interval for σ2 is ( ([n − 1]*S2 ) / (χ21−α/2,n−1 ) , ([n − 1]S2) / ( χ2α/2,n−1) , where χ2u,k denotes the 100×uth percentile of the χ2 distribution with k degrees of freedom, S2 = (n − 1)−1 * the sum of (Wi − Wbar )2 (from 1 to n) is the sample variance and Wbar = n−1 * the sum...

Let X1,...,Xn be Beta-distributed with α unkown and β = 3. Find the Method of Moments...

Let X1,...,Xn be Beta-distributed with α unkown and β = 3. Find the Method of Moments estimator of α.

Subjects