1. Suppose that P is the uniform distribution on [0,1). Partition the interval [0,1) into equivalence...
4. Suppose that Z є R2p has a MVN distribution Nop( 12. E.). Partition Z as Z where X є Rp and Y є Rp. Denote the means of X and Y as μι and μy, respectively. Let μΔ-Ha-ty. Suppose that we obtain IID data Z1, ,Zn from the underlying distribution of Z. Let a (0,1) be a constant (a) Describe how to construct a (1-a)-level convex confidence region (CR) for μΔ when y, is known. Explain. (b) Describe how...
Let r be any rational number and define L = { x in Q: x < r }, the set of rational numbers less than r. Show that L is a Dedekind cut by proving the following properties: A. There exists a rational number x in L and there exists a rational number y not in L. ( This proves L is nonempty and L is not equal to Q) B. If x in L, then there exists z in...
A function f : Rn λε [0,1] R is strictly convex if for all x, y є Rn and all fax + (1-λ)y) < λ/(x) + (1-1)f(y) A symmetric matrix P-AT +A is called positive-definite if all its eigenvalues are positive. Show that a quadratic function f(x) -xPx is a convex function if and only P is positive-definite. A function f : Rn λε [0,1] R is strictly convex if for all x, y є Rn and all fax +...
Question 5: Consider the following choice (where pe 0,1), (0,1), and p+q<1): x=($40.p: $10,4; $0,1-p-9) vs. y= (a) For what values of p and does lottery x dominate lottery y? (b) For what values of p and q does lottery y dominate lottery x?
1. Let A= {0,1}2 U... U{0,1}5 and let < be the order on A defined by (s, t) E< if and only if s is a prefix of t. (We consider a word to be a prefix of itself.) (a) Find all minimal elements in A. (Recall that an element x is minimal if there does not exist y E A with y < x.) (b) Are 010 and 01101 comparable? 2. Give an example of a total order on...
Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that f is the PDF of some continuous random variable of interest, that F is the corresponding CDF, and assume that F is invertible (so that the function F-1 exists and gives a unique value). Show that the random variable X = F-1(U) has PDF f(x)—that is, that X has the desired PDF. Hint: use results on transformations of random variables. This cute result allows...
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an Suppose f is a continuous and differentiable function on...
can you please prove the following theorem using the provided axioms and defintions. using terms like suppose in a paragraph format. please write clearly or type if you can ! 1 Order Properties Undefined Terms: The word "point and the expression "the point z precedes the point y will not be defined. This undefined expression wil be written z < y. Its negation, "z does not precede y," will be written y. There is a set of all points, called...
Suppose f : B(0.1) C is holomorphic, with irg:) 1 for every z є B(0,1). Suppose also that f(0)-0, so f(z)g(2) for some holomorphic function g: B(0,1)C. (a) By applying the Maximum Principle to g on B(0, r) where 0 < r < 1 , deduce that If( S for every 2E (0, 1) . (b) Show also that |f'(0) S1 (c) Show that if lf(z)- for some z B(0,1)\(0), or if If,(0)| = 1 , then there is a...
2. [14 marks] Rational Numbers The rational numbers, usually denoted Q are the set {n E R 3p, q ZAq&0An= Note that we've relaxed the requirement from class that gcd(p, q) = 1. (a) Prove that the sum of two rational numbers is also a rational number (b) Prove that the product of two rational numbers is also a rational number (c) Suppose f R R and f(x)= x2 +x + 1. Show that Vx e R xe Qf(x) Q...