lecture17.pdf - COMPSCI 240 Reasoning Under Uncertainty...

Info icon This preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon
COMPSCI 240: Reasoning Under Uncertainty Arya Mazumdar University of Massachusetts at Amherst Fall 2016
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Lecture 17 Review Class
Image of page 2
Multiple Random Variables
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Multiple Random Variables Consider two random variables, X and Y mapping from Ω to R .
Image of page 4
Multiple Random Variables Consider two random variables, X and Y mapping from Ω to R . The probabilities of these events give the joint PMF of X and Y : P ( X = i , Y = j ) = P ( { X = i , Y = j } ) = p X , Y ( i , j )
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Multiple Random Variables Consider two random variables, X and Y mapping from Ω to R . The probabilities of these events give the joint PMF of X and Y : P ( X = i , Y = j ) = P ( { X = i , Y = j } ) = p X , Y ( i , j ) Two discrete random variables X and Y are independent if and only if P ( X = a , Y = b ) = P ( X = a ) P ( Y = b ) for all a and b .
Image of page 6
Conditioning Conditional PMF of X given Y : P ( X = i | Y = j ) = P ( { X = i }|{ Y = j } ) .
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Conditioning Conditional PMF of X given Y : P ( X = i | Y = j ) = P ( { X = i }|{ Y = j } ) . Compute P ( X | Y ) using the definition of conditional probability: P ( X = i | Y = j ) = P ( X = i , Y = j ) P ( Y = j ) since for any two events A , B we have P ( A | B ) = P ( A B ) P ( B ) .
Image of page 8
Conditioning Conditional PMF of X given Y : P ( X = i | Y = j ) = P ( { X = i }|{ Y = j } ) . Compute P ( X | Y ) using the definition of conditional probability: P ( X = i | Y = j ) = P ( X = i , Y = j ) P ( Y = j ) since for any two events A , B we have P ( A | B ) = P ( A B ) P ( B ) . The conditional probability P ( X = i | Y = j ) is the joint probability P ( X = i , Y = j ) normalized by the marginal P ( Y = j ).
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Conditioning Conditional PMF of X given Y : P ( X = i | Y = j ) = P ( { X = i }|{ Y = j } ) . Compute P ( X | Y ) using the definition of conditional probability: P ( X = i | Y = j ) = P ( X = i , Y = j ) P ( Y = j ) since for any two events A , B we have P ( A | B ) = P ( A B ) P ( B ) . The conditional probability P ( X = i | Y = j ) is the joint probability P ( X = i , Y = j ) normalized by the marginal P ( Y = j ). An equivalent definition of independence is X and Y are independent if for all i , j , P ( X = i | Y = j ) = P ( X = i )
Image of page 10
Suppose a student takes a Moodle quiz containing two questions. Let Q be the number of questions she gets correct. Let S be the number of hours she had studied that week. Suppose S takes the value 0, 1, or 2. If we pick a random student from the class, the joint probabilities of Q and S are as follows: S = 0 S = 1 S = 2 Q = 0 0.1 0.1 0 Q = 1 0.1 0.2 0 Q = 2 0 0 0.5 So, for example, P ( Q = 1 , S = 1) = 0 . 2. What will be P ( Q = 1) =?; P ( Q = 2) =?; P ( S = 1) =?; P ( S = 2) =?
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Suppose a student takes a Moodle quiz containing two questions. Let Q be the number of questions she gets correct. Let S be the number of hours she had studied that week. Suppose S takes the value 0, 1, or 2. If we pick a random student from the class, the joint probabilities of Q and S are as follows: S = 0 S = 1 S = 2 Q = 0 0.1 0.1 0 Q = 1 0.1 0.2 0 Q = 2 0 0 0.5 So, for example, P ( Q = 1 , S = 1) = 0 . 2. What will be P ( Q = 1) =?; P ( Q = 2) =?; P ( S = 1) =?; P ( S = 2) =? Are Q and S independent? Justify your answer.
Image of page 12
Suppose a student takes a Moodle quiz containing two questions.
Image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 14
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

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