Develop a careful proof of Theoram 2.1 which states that for any events A and B, Pr

Question:

Develop a careful proof of Theoram 2.1 which states that for any events A and B,

Pr (A U B) = Pr (A) + Pr (B) – Pr (A ∩ B).

One way to approach this proof is to start by showing that the set can be written as the union of three mutually exclusive sets,

And hence by Corollary 2.1,

Next, Show that

And likewise

Put these results together to complete the desired proof.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Probability and Random Processes With Applications to Signal Processing and Communications

ISBN: 978-0123869814

2nd edition

Authors: Scott Miller, Donald Childers

See More Books

Question Posted: Nov 19, 2015 11:52 PM

See More Questions

Formally prove the union bound which states that for any events,A1, A2...A3, ( not necessarily mutually exclusive), Pr(A;) Pr U) Pr(A.) i=1 i=1
Prove Jensens inequality, which states that for any convex function g (x) and any random variable X, E [ g( X)] g ( E [ X]).
The probability of a type II error for the t test for H0: Î²1 = Î²10 can be computed in the same manner as it was computed for the t tests of Chapter 8. If the alternative value...
In a survey of 106 college students, data were gathered on the number of piercings each student had and the number of classes they had skipped in the past 3 weeks. We want to determine whether there...
Cecillia Corporation is preparing the comparative financial statements to be included in the annual report to shareholders. Cecillia's fiscal year ends May 31. The following information is available....
Helix Corporation uses the weighted average method in its process costing system. It produces prefabricated flooring in a series of steps carried out in production departments. All of the material...
P 5-8 Consolidated workpapers (downstream sales) Pop Corporation acquired 100 percent of Son Corporations outstanding voting common stock on January 1, 2016, for $660,000 cash. Sons stockholders...
Two items are omitted in each of the following four lists of income statement data. Determine the amounts of the missing items, identifying them byletter. igl Sas0,000 Sales Seles returns and...
Waterways is thinking of mass-producing one of its special-order sprinklers. To do so would increase variable costs for all sprinklers by an average of $0.70 per unit. The company also estimates that...
Highway Driving School charges $800 per student to prepare and administer written and driving tests. Variable costs of $240 per student include trainers' wages, study materials, and gasoline. Annual...
This classic problem was given widespread pop culture attention in 2008 by the movie "21." A form of this problem known as the Monty Hall problem appeared in the "Ask Marilyn" column in Parade...
Show that the above formula for the probability of the union of two events can be generalized to three events as follows: Pr (A U B U C) = Pr (A) + Pr (B) + Pr(C) Pr( A B) Pr( A C) Pr( B C)+ Pr( A B...
Distinguish between CRM in the B2C world and CRM in the B2B world.
State the vertical asymptotes, if any exist for the function. T f(x) = x+81
4. Oh no! Prof. Conlin was doing the dishes, but ran out of space on his drying rack. He decided to set the last two bowls on a towel on the counter to dry. He wondered, "To make sure they get dry,...
It has been assumed so far that the firm will operate a project over its full physical life. However, may not be the best option - it may be better to abandon a project prior to the end of potential...
33-34 Find (a) f + g, (b) f- g, (c) fg, and (d) f/g and state their domains. 33. f(x)=25-x, g(x) = x+1 ===== 1 34. f(x)= x-1' 9(x)=-2 X
Describe five steps independent auditors take when auditing an organization. -An independent auditor, often a public accounting firm, begins an audit by studying the business. This approach helps to...
Z = (X, Y) f Z (x, y) = 4/5 ( 2 x y 3 ) for x , y [0, 1]. Find the marginal probabilities f x and f y , the covariance xy , and determine whether X and Y are independent.
2. In the circuit given in Figure 2, i,(t) = 5.67cos(5t)A and v (t) = 70.71 cos(5t 60) V a) Find the equivalent load impedance. State whether the load is inductive or capacitive. b) Calculate the...
$6 \times 23$ Use clock arithmetic $(\bmod 12)$, to perform the indicated calculation.
A California study concluded that by following 7 simple health rules a man's life can be extended by 11 years on the average and a woman's life by 7 years. These 7 rules are as follows: no smoking,...
A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different heating systems, a garage or carport, and a patio or screened porch. How many different, plans are...
A drug for the relief of asthma can be purchased from 5 different, manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. How many different ways can a...
nformation pertaining to Noskey Corporation s sales revenue follows: November 2 0 2 1 ( Actual ) December 2 0 2 1 ( Budgeted ) January 2 0 2 2 ( Budgeted ) Cash sales $ 1 0 5 , 0 0 0 $ 1 1 5 , 0 0 0...
The management team of Netflix maintains a stable dividend using the Lintner model: Dt+1 = Dt + EPS Target Payout Where Dt (Dt+1) = dividend in the current period t (the next period t + 1) EPSt =...
#1 #2 hapter 50 10 D Werences lav Help Required information [The following information applies to the questions displayed below) Archer Company is a wholesaler of custom-built air-conditioning units...

Develop a careful proof of Theoram 2.1 which states that for any events A and B, Pr

Question:

Pr. (A U B) = Pr (A n (ATB)) + Pr (A n B) + Pr (B n (A nB)) Pr (A n {A n B)) = Pr (A)-Pr (A n B) Pr (B n {A n B) = Pr (B)-Pr (A n B)

Step by Step Answer:

First note that the sets A UB and B can be rewritten as AUBAUB B UB A...View the full answer

Probability and Random Processes With Applications to Signal Processing and Communications

Students also viewed these Statistics questions