5. Consider an experiment in which the sample space Ω is precisely the real line 9t...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
(i) Show that 15 (ii) Show that (X) 5/12 and E(Y) 5/8 3(1 - 2X2 +X4) 4(2- 3X +X3) (iii) Show that 3(y|X) (iv) Verify thatE(Y)E(Y) 14] 7. (a) State Chebyshev's inequality and prove it using Markov's inequality 15] (b) Let (2, P) be a probability space representing a random experiment that can be repeated many times under the same conditions, and let A C S2 be a random event. Suppose the experiment is repeated n times (i) Write down...
5. Let Ω = { 1, 2, 3, . j be the countably infinite sample space whose elements (outcomes) are the positive integers. For each positive integer n, define the event An k k is a multiple of n \ (a) Find n and m such that An-Ag n A4 and Am-A6 A9. (b) If P((k]) -fnd the probability of the event As k-1 Note: an exact answer is required here; if you write a program to obtain answers of...
9. Consider the sample space Ω {1.2.3.4 } (the set of all natural numbers). We want to show that there is no probability measure on 2 under which "all outcomes are equally likely" Let's argue by contradiction. Suppose P is a probability measure such that P)) has the same value for all n e2. Let's see what can go wrong. (a) Suppose P)> 0. Which axiom of probability will be violated? (b) Suppose P((n)) = 0, which axiom of probability...
5. Consider the sample space Ω = [0, 1]. Let P be a probability function such that for any interval fa, b, P(a, b-b-a. In other words, probabilty of any interval is its length Let us start with Co [0, 1, and at nth step, we define Cn by removing an interval of length 1/3 from the middle of each interval in Cn-1 For example, C1-[0, 1/3 u [2/3,1], C2-[0,1/9)U[2/9,1/3 U [2/3,7/9 U[8/9, 1] and so on. Here is a...
2) Consider the sample space of three coin tosses: Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }. Assuming all elements to be equally likely, we assign P({ωi}) = 1/8, i = 1, 2, 3, 4, 5, 6, 7, 8. Define random variable to capture the second and third outcomes of the toss: X2 = { 0, if second outcome is T; 1, if second outcome is H and X3 = { 0, if third outcome is T;...
1. (10 marks) (a) Let m events Bi, , Bm form a partition of the sample space Ω and let event A be any event such that A c S2. Then show that given Bi > 0 for j = 1,.. ., m (b) Considcr a clinical trial where group of paticnts arc trcated for depression. As in many such trials a patient has two possible out- comes, in this study a relapse and no relapse. Refer to a relapse...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
Problem 5. Given a vector space V, a bilinear form on V is a function f : V x V -->R satisfying the following four conditions: f(u, wf(ū, ) + f(7,i) for every u, õ, wE V. f(u,ū+ i) = f(u, u) + f(ū, w) for every ā, v, w E V. f(ku, kf (ū, v) for every ū, uE V and for every k E R f(u, ku) = kf(u, u) for every u,uE V and for every k...
QUESTION 1 Consider the following OLS regression line (or sample regression function): wage =-2.10+ 0.50 educ (1), where wage is hourly wage, measured in dollars, and educ years of formal education. According to (1), a person with no education has a predicted hourly wage of [wagehat] dollars. (NOTE: Write your answer in number format, with 2 decimal places of precision level; do not write your answer as a fraction. Add a leading minus sign symbol, a leading zero and trailing...