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6.041 Lecture Notes - Probability Axioms, Sample Space, Negative Number

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arethelion
8 Oct 2022
School
MIT
Department
Electrical Engineering and Computer Science
Course
6.041
Professor

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Document Summary

Lecture notes from mit departments of electrical engineering and computer science. Pro b ab ility mo d els an d axio ms. Objective: introduction to the general framework of probability models, multiple discrete or continuous random variables. Definition: the sample space =list (set) of possible outcomes . Properties: mutuality exclusive, collectively exhaustive and at the "right" granularity. Family cares if the person is "ill", and public health service cares if the test is "positive or. It makes sense to model them as two discreet events: disease= ill/test = negative, disease= ill/=test positive. They are typically well named but contain very little information. An event that is too fine grained isn"t useful in and of itself, you have to combine it with lots of other events before you can decide what to do next. Roll of a die, sample space ( ) is [1, 2, 3, 4, 5, 6], discrete sample spaces. Example: representation of two rolls of a hexahedral die.

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Related Questions

Grade 9 Science electricity multiple choice:

  1. All of the following are examples of a load in an electric circuit, except…
  2. heaters b. motors c. lamps d. batteries

 

  1. A continuous path in which electrons can flow is called a(n)…
  2. electric circuit b. photovoltaic cell c. electric cell d. electrolyte

 

  1. The part of an electric circuit that converts electrical energy into other forms of energy is called a(n)…
  2. connecting wire b. energy source c. load d. switch

 

  1. What type of energy does a fan convert into motion?light energy b. thermal energy c. electrical energy d. none of the above

 

  1. An electric cell that may only be used once is called a… a. primary cell secondary cell c. fuel cell d. photovoltaic cell
  1. Electrons will only flow in an electrical circuit if …
  2. the circuit breaker is open. c. their path is complete and continuous
  3. there is a gap between the conductors d. the switch is not closed.

 

  1. The rate at which electric charges (electrons) move past a given point in a circuit is called…
  2. voltage b. resistance c. current d. none of the above

 

  1. The symbol for electric current is…
  2. I b. V c. A d. C

 

  1. A device used to measure potential difference is called a(n)…
  2. electroscope b. ohmmeter c. ammeter d. voltmeter

 

  1. Electrical resistance is measured in…

      a. amperes b. watts c. ohms d. volts

  1. Electrical Resistance is the ability (or force) that …
  2. stops or slows down the electric charges in a circuit        
  3. that can either help or stop the electric charges in a circuit
  4. helps or speeds up the electric charges in a circuit        
  5. none of these are correct
  1. Loads connected in series…
  2. do not affect the total resistance of the circuit       c. increase the total resistance of the circuit
  3. decrease the total resistance of the circuit       d. It depends on the wattage of the loads

 

  1. What will happen to other lamps on a parallel circuit if an additional light bulb is added?
  2. They will continue to glow the same brightness               c. Some will get brighter and some will go out
  3. They will all continue to glow and will get brighter            d. They will all go out

 

  1. A series circuit has three identical lamps. The potential difference of the energy source is 30V. The resistance of each lamp is 5 Ω. What is the current through each lamp? (Hint: draw a picture of the circuit)

 

  1. 0.167 A b. 2 A c. 6 A d. 150 A

 

  1. One watt of electrical power is equal to…
  2. one joule per second       b. one joule per minute   c. ten joules per second d. ten joules per minute

 

  1. Electrical energy in your home is measured in…
  2. Joules       b. Amps       c. Kilowatt-Hours     d. Watts

 

  1. If you wish to lower your energy costs, which of the following times would be best to use an electric clothes dryer?
  2. Monday at 1:00 p.m.   b. Tuesday at 9:00 a.m. c. Friday at 3:00 p.m.             d. Saturday at 8:00 p.m.

 

  1. An older electrical safety device that contains a small wire that melts and breaks apart when current is too high is a…
  2. Circuit Breaker       b. Fuse       c. Ground Fault Interrupter       d. 3-hole outlet

 

  1. This electrical safety device, which is typically found in bathrooms, sc

 

  1. Circuit Breaker       b. Fuse       c. Ground Fault Interrupter     d. 3-hole outlet

 

  1. Which of the following devices measures the amount of energy used every hour in a house or apartment building?
  2. Phantom load       b. Fuse               c. Smart Meter         d. Ground Fault Interrupter

 

Writing e Persuasive Essay: Step I - Prewrite 
1-2-3 Structure of a Persuasive Essay 
1. INTRODUCTION 
   - Hook 
   - Setting/Background 
   - Thesis Statement 
 2. SUPPORTING POINTS (at least 3)
   - #1Counterargument, Argument, Support 
   - #2 Counterargument, Argument, Support 
   - #3 Counterargument, Argument, Support
3. CONCLUSION 
   - Relates to thesis and ties supporting points together 
   - Provocative final statement 
 
Assignment: 
 Complete step 1 in writing a Persuasive Essay (see samples below): 
Step 1) Prewrite.
* Choose a topic. (Your topic should be one you feel strongly about, that has more than one 
 side, and that you can find evidence to support.) 
* Take a position. (Begin by taking a position, but know you can change it if your evidence 
changes your thinking.) 
 * Write a simple thesis statement. (This should include your topic, your stance, and a list of 
possible supporting points.) 
* Research the topic if necessary. (Take careful notes, gather quotes and details in support of
 your stance, and accurately record your sources.) 
* Make an Outline using the 1-2-3 Structure above (Sample on Page 3)
 Introduction - Include a  thesis statement - (your topic, your stance, and a brief summary of 
 your supporting points). 
- Supporting points - List main points for supporting paragraphs (Include
 corresponding arguments and counterarguments. NOTE: Present counter arguments first for a 
stronger essay). 
- Conclusion - Relate your conclusion to your thesis, and use it to tie your supporting points together. 
 
 
            SAMPLE PERSUASIVE ESSAY
SAMPLE: Prewrite
TOPIC: Space debris
POSITION: We need a definitive global policy on cleaning up existing space debris and preventing it in the future.
THESIS STATEMENT: Due to the increasing risk of space debris falling back to earth in inhabited areas or damaging orbiting spacecraft, we, as a global community, need to define and adopt a comprehensive policy that eliminates existing space debris and prevents it in the future.
RESEARCH: NASA and other government sources plus newspaper articles. Checked data and quotes, and noted sources to give proper credit.]
 
SAMPLE: Outline
THESIS STATEMENT: Due to the increasing risk of space debris falling back to earth in inhabited areas or damaging orbiting spacecraft, we, as a global community, need to define and adopt a comprehensive policy that eliminates existing space debris and prevents it in the future.
LIST OF POINTS, COUNTERARGUMENTS, AND ARGUMENTS - by paragraph
1) POINT: Space debris threatens humans on earth, Space Station
    COUNTER: Chance is very small: Very few people have been hit
    ARGUMENT: Recent threats to Space Station required intervention
2) POINT: Space debris threatens our communications satellites
    COUNTER: There is very little damage percentage-wise; we can fly satellites to avoid debris
    ARGUMENT: Problem going to get worse; soon can't avoid collisions
3) POINT: Tracking space debris costs a huge amount of time and money, we need a plan to prevent debris in the first place
    COUNTER: Making plan will cost a lot of time and money
    ARGUMENT: Plan will cost less than inevitable damage.
CONCLUSION: Although the chance of humans being hit or satellites being damaged by space debris is small at this time, and the cost of preventing debris would be substantial, the space debris problem is only going to grow. Space debris will become a material threat at some point in the not-too-distant future. The best solution is to take it on now with a global plan.
 

*Please refer to the attached file.

You are writing a rough draft for the essay part 1,

This writing practice will focus on

Steps *1 and *2 below:

___________________

~

FIVE STEPS OF WRITING~*1- Prewrite*2-Draft 3- Revise 4- Edit/Proofread 5- Publish

Copyright © Acellus Corporation. All Rights Reserved. www.acellus.com

Expository Essay Part 1

Student Resource

EXPOSITORY ESSAY

– explains, points out

-

Used in articles, essays, exam questions

- Used to report on research

- Used to share information, give directions

REVIEW:

Elements of Expository Essay:

Thesis

–

Thesis Statement states topic, introduces supporting points

Structure

–

- Organization - often by Cause/Effect, Problem/Solution, Compare/Contrast

- Supporting paragraphs (at least one per supporting point)

- Use of formatting, graphics, and multi-media where appropriate

- Conclusion - relates to thesis and pulls supporting points together

Style

–

- Formal tone - complete sentences; conventional construction and language

- Active voice

Assignment:

Complete the first

two

steps in writing an Expository Essay.

You can use the sample writing prompt on Page 3 or choose a topic based on stories/articles of

your choosing. Page 3 also gives a sample thesis and outline based on the writing prompt. If

writing on your own topic and texts, choose three texts from different genres – e.g., short story,

science fiction, poetry, memoir, informational text, etc.

Step 1)

Prewrite

– Write your thesis statement.

– Determine the structure and organization (e.g., compare/contrast).

– Make an Outline (should include the thesis statement plus a list of

supporting points and a conclusion).

Step 2)

Draft

– Follow your Outline.

– Consider the audience's knowledge of and experience with the topic.

– Give concrete examples (each supporting point should be at least one

paragraph).

– Relate each point back to the thesis statement.

– Draw a conclusion that relates to the thesis statement and ties the

supporting points together.

Copyright © Acellus Corporation. All Rights Reserved. www.acellus.com

Expository Essay Part 1

Student Resource

EXPOSITORY ESSAY - SAMPLE THESIS AND OUTLINE

WRITING PROMPT

:

Compare/contrast the authors' treatments of the theme of

taking

responsibility

in three different genres of text:

Short Story

–"Through the Tunnel,"

Science

Fiction

– "A Sound of Thunder," and

Informational Text

– "Quit blaming; take responsibility."

*

Thesis

:

Authors Lessing, Bradbury, and Gitomer, writing in three different textual genres,

each address the theme of taking responsibility by contrasting instances of responsible and

irresponsible behavior and drawing the conclusion of the benefits of taking responsibility.

*

Supporting paragraphs

– with formatting (subtitles):

1. Short Story

– Lessing ("Through the Tunnel") contrasts Jerry's irresponsible/ responsible

behavior and draws conclusion responsibility is best – Jerry feels better about himself.

- Irresponsible: Jerry tries to get attention, be accepted by other boys; wants to swim

through tunnel before ready (examples from story)

- Responsible: Jerry stops trying to please other boys; adopts his quest; trains and works

to be prepared (examples from story)

- Lessing draws unstated conclusion of benefits of taking responsibility

(contrasts Jerry's feelings before and after quest)

2. Science Fiction

– Bradbury ("A Sound of Thunder") contrasts Time Safari Inc.'s/ Eckels'

responsible/irresponsible behavior and graphically illustrates the devastating results of

irresponsibility.

- Irresponsible: Eckels' uncaring, irresponsible attitude (examples in story)

- Responsible: Time Safari's careful planning and preparation for the hunt

(examples

from story)

- Bradbury ends story showing dramatic impact of being irresponsible, implying the

benefits of responsibility

3. Informational Text

–

Gitomer ("Quit blaming; take responsibility") compares salespeople

with politicians in their tendency to avoid taking responsibility and contrasts that attitude with

suggestions on taking responsibility to achieve success.

- Irresponsible: Salespeople and politicians blame others for their failures

(specific examples from article)

- Responsible: Suggestions on how they can take responsibility by

studying failures and

making them successes (examples in article)

- Gitomer concludes that salespeople (and politicians) can achieve results

they seek by

overcoming obstacles and taking responsibility

*

Conclusion

:

Authors Lessing, Bradbury, and Gitomer, writing in three different textual

genres, each address the theme of taking responsibility by contrasting instances of responsible

and irresponsible behavior, concluding that taking responsibility brings the best r

This writing practice will focus on

Steps *1 and *2 below:

___________________

~

FIVE STEPS OF WRITING ~* 1-Prewrite* 2- Draft 3- Revise\ 4- Edit/Proofread 5- Publish

Copyright © Acellus Corporation. All Rights Reserved. www.acellus.com

Expository Essay Part 1

Student Resource

EXPOSITORY ESSAY

– explains, points out

-

Used in articles, essays, exam questions

- Used to report on research

- Used to share information, give directions

REVIEW:

Elements of Expository Essay:

Thesis

–

Thesis Statement states topic, introduces supporting points

Structure

–

- Organization - often by Cause/Effect, Problem/Solution, Compare/Contrast

- Supporting paragraphs (at least one per supporting point)

- Use of formatting, graphics, and multi-media where appropriate

- Conclusion - relates to thesis and pulls supporting points together

Style

–

- Formal tone - complete sentences; conventional construction and language

- Active voice

Assignment:

Complete the first

two

steps in writing an Expository Essay.

You can use the sample writing prompt on Page 3 or choose a topic based on stories/articles of

your choosing. Page 3 also gives a sample thesis and outline based on the writing prompt. If

writing on your own topic and texts, choose three texts from different genres – e.g., short story,

science fiction, poetry, memoir, informational text, etc.

Step 1)

Prewrite

– Write your thesis statement.

– Determine the structure and organization (e.g., compare/contrast).

– Make an Outline (should include the thesis statement plus a list of

supporting points and a conclusion).

Step 2)

Draft

– Follow your Outline.

– Consider the audience's knowledge of and experience with the topic.

– Give concrete examples (each supporting point should be at least one

paragraph).

– Relate each point back to the thesis statement.

– Draw a conclusion that relates to the thesis statement and ties the

supporting points together.

Copyright © Acellus Corporation. All Rights Reserved. www.acellus.com

Expository Essay Part 1

Student Resource

EXPOSITORY ESSAY - SAMPLE THESIS AND OUTLINE

WRITING PROMPT

:

Compare/contrast the authors' treatments of the theme of

taking

responsibility

in three different genres of text:

Short Story

–"Through the Tunnel,"

Science

Fiction

– "A Sound of Thunder," and

Informational Text

– "Quit blaming; take responsibility."

*

Thesis

:

Authors Lessing, Bradbury, and Gitomer, writing in three different textual genres,

each address the theme of taking responsibility by contrasting instances of responsible and

irresponsible behavior and drawing the conclusion of the benefits of taking responsibility.

*

Supporting paragraphs

– with formatting (subtitles):

1. Short Story

– Lessing ("Through the Tunnel") contrasts Jerry's irresponsible/ responsible

behavior and draws conclusion responsibility is best – Jerry feels better about himself.

- Irresponsible: Jerry tries to get attention, be accepted by other boys; wants to swim

through tunnel before ready (examples from story)

- Responsible: Jerry stops trying to please other boys; adopts his quest; trains and works

to be prepared (examples from story)

- Lessing draws unstated conclusion of benefits of taking responsibility

(contrasts Jerry's feelings before and after quest)

2. Science Fiction

– Bradbury ("A Sound of Thunder") contrasts Time Safari Inc.'s/ Eckels'

responsible/irresponsible behavior and graphically illustrates the devastating results of

irresponsibility.

- Irresponsible: Eckels' uncaring, irresponsible attitude (examples in story)

- Responsible: Time Safari's careful planning and preparation for the hunt

(examples

from story)

- Bradbury ends story showing dramatic impact of being irresponsible, implying the

benefits of responsibility

3. Informational Text

–

Gitomer ("Quit blaming; take responsibility") compares salespeople

with politicians in their tendency to avoid taking responsibility and contrasts that attitude with

suggestions on taking responsibility to achieve success.

- Irresponsible: Salespeople and politicians blame others for their failures

(specific examples from article)

- Responsible: Suggestions on how they can take responsibility by

studying failures and

making them successes (examples in article)

- Gitomer concludes that salespeople (and politicians) can achieve results

they seek by

overcoming obstacles and taking responsibility

*

Conclusion

:

Authors Lessing, Bradbury, and Gitomer, writing in three different textual

genres, each address the theme of taking responsibility by contrasting instances of responsible

and irresponsible behavior, concluding that taking responsibility brings the best r