CMPSCI 240: Reasoning Under Uncertainty

Transcription

1 CMPSCI 240: Reasoning Under Uncertainty Lecture 8 Prof. Hanna Wallach wallach@cs.umass.edu February 16, 2012

2 Reminders Check the course website: ~wallach/courses/s12/cmpsci240/ Second homework is due TOMORROW

3 Recap

4 Last Time: Expectation PMF: p X (x) = P(X =x) = P({X =x})

5 Last Time: Expectation PMF: p X (x) = P(X =x) = P({X =x}) Expectation: E[X ] = x x p X (x)

6 Last Time: Expectation PMF: p X (x) = P(X =x) = P({X =x}) Expectation: E[X ] = x x p X (x) If Y = g(x ) then E[Y ] = x g(x) p X (x)

7 Last Time: Expectation PMF: p X (x) = P(X =x) = P({X =x}) Expectation: E[X ] = x x p X (x) If Y = g(x ) then E[Y ] = x g(x) p X (x) If Y = a X + b then E[Y ] = a E[X ] + b

8 Last Time: Variance Variance: var(x ) = E[(X E[X ]) 2 ] = E[X 2 ] E[X ] 2

9 Last Time: Variance Variance: var(x ) = E[(X E[X ]) 2 ] = E[X 2 ] E[X ] 2 If Y = a X + b then var(y ) = a 2 var(x )

10 Last Time: Variance Variance: var(x ) = E[(X E[X ]) 2 ] = E[X 2 ] E[X ] 2 If Y = a X + b then var(y ) = a 2 var(x ) Standard deviation: σ X = var(x )

11 Last Time: Variance Variance: var(x ) = E[(X E[X ]) 2 ] = E[X 2 ] E[X ] 2 If Y = a X + b then var(y ) = a 2 var(x ) Standard deviation: σ X = var(x ) If Y = a X + b then σ Y = a σ X

12 Last Time: Multiple Random Variables Joint PMF: p X,Y (x, y) = P({X =x} {Y =y})

13 Last Time: Multiple Random Variables Joint PMF: p X,Y (x, y) = P({X =x} {Y =y}) p X (x) = y p X,Y (x, y) and p Y (y) = x p X,Y (x, y)

14 Last Time: Multiple Random Variables Joint PMF: p X,Y (x, y) = P({X =x} {Y =y}) p X (x) = y p X,Y (x, y) and p Y (y) = x p X,Y (x, y) If Z = g(x, Y ) then E[Z] = x y g(x, y) p X,Y (x, y)

15 Last Time: Multiple Random Variables Joint PMF: p X,Y (x, y) = P({X =x} {Y =y}) p X (x) = y p X,Y (x, y) and p Y (y) = x p X,Y (x, y) If Z = g(x, Y ) then E[Z] = x y g(x, y) p X,Y (x, y) If Z = ax + by + c then E[Z] = a E[X ] + b E[Y ] + c

16 Multiple Random Variables (Cont.)

17 Three or More Random Variables The joint PMF of X, Y, and Z is denoted p X,Y,Z where p X,Y,Z (x, y, z) = P({X =x} {Y =y} {Z =z})

18 Three or More Random Variables The joint PMF of X, Y, and Z is denoted p X,Y,Z where p X,Y,Z (x, y, z) = P({X =x} {Y =y} {Z =z}) The expected value of g(x, Y, Z) is E[g(X, Y, Z)] = g(x, y, z) P X,Y,Z (x, y, z) x y z

19 Three or More Random Variables The joint PMF of X, Y, and Z is denoted p X,Y,Z where p X,Y,Z (x, y, z) = P({X =x} {Y =y} {Z =z}) The expected value of g(x, Y, Z) is E[g(X, Y, Z)] = x g(x, y, z) P X,Y,Z (x, y, z) y z E[aX + by + cz + d] = a E[X ] + b E[Y ] + c E[Z] + d

20 Linearity of Expectation In general, the expectation of a sum of random variables is equal to the sum of their expectations, i.e., E[X X n ] = n E[X i ] i=1

21 Linearity of Expectation In general, the expectation of a sum of random variables is equal to the sum of their expectations, i.e., E[X X n ] = n E[X i ] i=1 If X i has the same PMF as X j they are identically distributed: E[X i ] = x x p Xi (x) = x x p Xj (x) = E[X j ]

22 Examples of Three or More Random Variables e.g., an office party decides to do a gift exchange for their annual holiday party. Everyone in the office purchases a gift and places it in a box. Each person is then given a random gift drawn from the box. What is the expected number of people who get back the gift that they purchased?

23 Expectations of Standard Random Variables e.g., if X is a binomial random variable, what is E[X ]?

24 Conditioning

25 Conditioning Conditional PMF of X given Y : denoted by p X Y, where p X Y (x y) = P(X =x Y =y) = P({X =x} {Y =y})

26 Conditioning Conditional PMF of X given Y : denoted by p X Y, where p X Y (x y) = P(X =x Y =y) = P({X =x} {Y =y}) Compute p X Y using the definition of conditional probability: P(A B) = P(A B) P(B) = p X Y (x y) = p X,Y (x, y) p Y (y)

27 Conditioning and the Tabular Representation y 1 y 2 y 3 y 4 x x x Compute conditional PMF for Y given X = x by renormalizing the values in the row for x and conditional PMF for X given Y =y by renormalizing the values in the column for y

28 Conditioning and the Tabular Representation y 1 y 2 y 3 y 4 x x x e.g., to compute p X Y (x 1 y 2 ): p X Y (x 1 y 2 ) = p X,Y (x 1, y 2 ) 3 i=1 p X,Y (x i, y 2 ) = p X,Y (x 1, y 2 ) p Y (y 2 )

29 Calculating Joint and Marginal PMFs Apply the multiplication rule using the underlying events {X =x} {Y =y}, {X =x}, and {Y =y} to give p X,Y (x, y) = p X (x) p Y X (y x) = p Y (y) p X Y (x y)

30 Calculating Joint and Marginal PMFs Apply the multiplication rule using the underlying events {X =x} {Y =y}, {X =x}, and {Y =y} to give p X,Y (x, y) = p X (x) p Y X (y x) = p Y (y) p X Y (x y) Divide-and-conquer approach to calculating marginal PMFs: p X (x) = y p X,Y (x, y) = y p Y (y) p X Y (x y)

31 Examples of Conditioning e.g., if my computer has a virus (which is true with probability 0.2) my antivirus program will detect this virus with probability 0.7. If my computer does not have a virus, my antivirus program will mistakenly detect a virus with probability 0.1. Let X and Y be binary random variables that are equal to 1 if my computer has a virus and the program detects a virus, respectively, and 0 otherwise. What is p X,Y?

32 Conditioning on Events We can condition random variables on events too

33 Conditioning on Events We can condition random variables on events too If X is associated with the same experiment as event A then p X A (k) = P(X =k A) = P({X =k} A)

34 Conditioning on Events We can condition random variables on events too If X is associated with the same experiment as event A then p X A (k) = P(X =k A) = P({X =k} A) e.g., X = sum of two rolls, A = first roll is odd

35 Conditioning on Events We can condition random variables on events too If X is associated with the same experiment as event A then p X A (k) = P(X =k A) = P({X =k} A) e.g., X = sum of two rolls, A = first roll is odd Nothing fancy here this is just notation when in doubt, think about the underlying events and outcomes

36 Examples of Conditioning on Events e.g., a student will take a test repeatedly up to a maximum of n times, each time with a probability p of passing, independent of previous attempts. What is the PMF of the number of attempts, given that the student passes the test?

37 Conditional Expectation Conditional version of expectation: E[X Y =y] = x x p X Y (x y)

38 Conditional Expectation Conditional version of expectation: E[X Y =y] = x x p X Y (x y) For any function g( ) of X, what is E[g(X ) Y =y]?

39 Conditional Expectation Conditional version of expectation: E[X Y =y] = x x p X Y (x y) For any function g( ) of X, what is E[g(X ) Y =y]? E[g(X ) Y =y] = x g(x) p X Y (x y)

40 Conditional Expectation Conditional version of expectation: E[X Y =y] = x x p X Y (x y) For any function g( ) of X, what is E[g(X ) Y =y]? E[g(X ) Y =y] = x g(x) p X Y (x y) Expectations can be conditioned on events too

41 Conditional Variance and Standard Deviation Conditional variance of X given Y =y: var(x Y =y) = E[X 2 Y =y] E[X Y =y] 2

42 Conditional Variance and Standard Deviation Conditional variance of X given Y =y: var(x Y =y) = E[X 2 Y =y] E[X Y =y] 2 What is the conditional standard deviation of X given Y =y?

43 Conditional Variance and Standard Deviation Conditional variance of X given Y =y: var(x Y =y) = E[X 2 Y =y] E[X Y =y] 2 What is the conditional standard deviation of X given Y =y? σ X Y=y = var(x Y =y)

44 Conditional Variance and Standard Deviation Conditional variance of X given Y =y: var(x Y =y) = E[X 2 Y =y] E[X Y =y] 2 What is the conditional standard deviation of X given Y =y? σ X Y=y = var(x Y =y) Both can be conditioned on events too

45 Total Expectation Theorem Total expectation theorem: for any random variables X and Y E[X ] = y p Y (y) E[X Y =y]

46 Total Expectation Theorem Total expectation theorem: for any random variables X and Y E[X ] = y p Y (y) E[X Y =y] For any disjoint events A 1,..., A n that partition Ω: E[X ] = n P(A i ) E[X A i ] i=1

47 Expectations of Standard Random Variables e.g., if X is a geometric random variable, what is E[X ]?

48 Independence

49 Independence of Events Two events A and B are independent if and only if P(A B) = P(A) P(B)

50 Independence of Events Two events A and B are independent if and only if P(A B) = P(A) P(B) If P(B) > 0 this is equivalent to P(A B) P(B) = P(A B) = P(A)

51 Independence of Events Two events A and B are independent if and only if P(A B) = P(A) P(B) If P(B) > 0 this is equivalent to P(A B) P(B) = P(A B) = P(A) If P(A) > 0 this is equivalent to P(B A) = P(B)

52 Independence of Random Variables X and Y are independent if and only if for all x and y p X,Y (x, y) = p X (x) p Y (y)

53 Independence of Random Variables X and Y are independent if and only if for all x and y p X,Y (x, y) = p X (x) p Y (y) Equivalently if for all x and y such that p Y (y) > 0 p X,Y (x, y) p Y (y) = p X Y (x y) = p X (x)

54 Independence of Random Variables X and Y are independent if and only if for all x and y p X,Y (x, y) = p X (x) p Y (y) Equivalently if for all x and y such that p Y (y) > 0 p X,Y (x, y) p Y (y) = p X Y (x y) = p X (x) Or if p Y X (y x) = p Y (y) for all x and y such that p X (x) > 0

55 Independence and the Tabular Representation y 1 y 2 y 3 y 4 x x x Are X and Y independent? How can we tell?

56 Expectation and Independence If X and Y are independent then E[XY ] = x y p X,Y (x, y) x y = x y p X (x) p Y (y) = E[X ] E[Y ] x y

57 Expectation and Independence If X and Y are independent then E[XY ] = x y p X,Y (x, y) x y = x y p X (x) p Y (y) = E[X ] E[Y ] x y This is only true if X and Y are independent

58 Independence and Three or More Random Variables X, Y, and Z are independent if and only if for all x, y, and z p X,Y,Z (x, y, z) = p X (x) p Y (y) p Z (z)

59 Independence and Three or More Random Variables X, Y, and Z are independent if and only if for all x, y, and z p X,Y,Z (x, y, z) = p X (x) p Y (y) p Z (z) If random variables X, Y, and Z are independent then so are random variables of the form f (X ), g(y ), and h(z)

60 Variance and Independence If X 1, X 2,..., X n are independent then var(x 1 + X X n ) = n var(x i ) i=1

61 Variance and Independence If X 1, X 2,..., X n are independent then var(x 1 + X X n ) = n var(x i ) i=1 If X 1, X 2,..., X n are independent and identically distributed, then var(x 1 ) = var(x 2 ) =... = var(x n ) and var(x 1 + X X n ) = n var(x 1 )

62 For Next Time Check the course website: ~wallach/courses/s12/cmpsci240/ Second homework is due TOMORROW