I go through the following process Close my eyes and pick a card Pick a side at

I go through the following process close my eyes and

This preview shows page 15 - 24 out of 70 pages.

I go through the following process. Close my eyes and pick a card Pick a side at random Show you that side Suppose I show you red. What is the probability the other side is red too? Let X 1 be the random side of the random card I chose Let X 2 be the other side of that card Compute P ( X 2 = red | X 1 = red ) P ( X 2 = red | X 1 = red ) = P ( X 1 = R , X 2 = R ) P ( X 1 = R ) COS 424/SML 302 Probability and Statistics Review February 6, 2019 14 / 69
Image of page 15

Subscribe to view the full document.

Now we can solve the card problem Now we can solve the card problem. Let X 1 be the random side of the random card I chose Let X 2 be the other side of that card Compute P ( X 2 = red | X 1 = red ) P ( X 2 = red | X 1 = red ) = P ( X 1 = R , X 2 = R ) P ( X 1 = R ) Numerator is 1/3: Only one card has two red sides. Denominator is 1/2: Three sides out of six are red. So P ( X 2 = red | X 1 = red ) = 2 / 3 Is it possible for a conditional probability to be outside of [0 , 1]? COS 424/SML 302 Probability and Statistics Review February 6, 2019 15 / 69
Image of page 16
Gender bias at Berkeley: Conditional probabilities From the textbook Statistics by Freedman, Pisani, & Purves: These numbers show that the system is biased ( p = 1 . 333 × 10 - 10 ) COS 424/SML 302 Probability and Statistics Review February 6, 2019 16 / 69
Image of page 17

Subscribe to view the full document.

Gender bias at Berkeley: Simpson’s Paradox From the textbook Statistics by Freedman, Pisani, & Purves: When the graduate school asked each of the departments to make changes in their admissions policy, all of the departments noted that none of their admissions numbers were biased against women. What important variable are we not incorporating in this comparison? COS 424/SML 302 Probability and Statistics Review February 6, 2019 17 / 69
Image of page 18
Gender bias at Berkeley: Condition on department From the textbook Statistics by Freedman, Pisani, & Purves: Conditioning on the department, the bias is reversed. Why? COS 424/SML 302 Probability and Statistics Review February 6, 2019 18 / 69
Image of page 19

Subscribe to view the full document.

Gender bias at Berkeley: Condition on department From the textbook Statistics by Freedman, Pisani, & Purves: Women are more likely than men to apply to departments with low admission rates. COS 424/SML 302 Probability and Statistics Review February 6, 2019 19 / 69
Image of page 20
The chain rule The definition of conditional probability lets us derive the chain rule The chain rule factorizes a joint distribution as a product of conditional distributions: P ( X , Y ) = P ( X , Y ) P ( Y ) P ( Y ) = P ( X | Y ) P ( Y ) Example: use conditional and marginal to get joint distribution Let Y be a disease and X be a symptom. We may know P ( X | Y ) and P ( Y ) from data. Use chain rule to find the probability of the disease and the symptom. COS 424/SML 302 Probability and Statistics Review February 6, 2019 20 / 69
Image of page 21

Subscribe to view the full document.

The chain rule across n variables In general, for any set of n variables P ( X 1 , . . . , X n ) = P ( X 1 ) n Y i =2 P ( X i | X 1 , . . . , X i - 1 ) COS 424/SML 302 Probability and Statistics Review February 6, 2019 21 / 69
Image of page 22
Marginalization Given a collection of RVs, we might only consider a subset of them.
Image of page 23

Subscribe to view the full document.

Image of page 24
  • Spring '09
  • Probability theory

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes