Intro to Probability Instructor: Alexandre Bouchard

Transcription

1 Intro to Probability Instructor: Alexandre Bouchard

2 Announcements Webwork out Graded midterm available after lecture

3 Regrading policy IF you would like a partial regrading, you should, BEFORE or ON Friday March 15, hand in to me at the beginning of a lecture: your exam a clean piece of paper stapled to it that clearly (i) explains the question(s) you would like us to regrade AND (ii) the issue(s) you would like to raise NOTE: for fairness, the new grade for the question could stay the same, increase, or, in certain cases, decrease

4 Plan for today Multivariate distributions, continued Independence of continuous random variables

5 Review: joint and marginal densities

6 Def 23 Today: density (for two random variables) The function f(x, y) is a joint density for X, Y if for any subset A of the plane: Z P ((X, Y ) 2 A) = (x,y)2a f(x, y) dx dy volume = probability Example: A = [a, b] x [c, d] height = density Notation for rectangle with one side equal to [a, b] and the other equal to [c,d] x b a c d y

7 Ex 60 Example: uniform density on a subset B of the plane Example: B y density x x height = density = 1/ area(b) y f(x, y) = 1 B(x, y) area(b) Recall: 1 B (x, y) = 1 if (x, y) 2 B 0 o.w.

8 Ex 59 Motivating problem A man and a woman try to meet at a certain place between 1:00pm and 2:00pm. Suppose each person pick an arrival time between 1:00pm and 2:00pm uniformly at random, and waits for the other at most 10 minutes. What is the probability that they meet?

9 Def 24 Example of marginal densities Marginal of X 0.6 Marginal fx(x) density x of Y fy(y) Height of the marginal at x = 0 obtained by integrating the joint density over y at x = 0: Y 0.0 y 0.0 f X (x) = Z +1 1 f(x, y) dy X density

10 Independence vs. dependence for continuous random variables

11 Def 25 Equivalent definitions X and Y are independent Useful to show that r.v. s are NOT indep For all intervals, A1, A2: P (X 2 A 1,Y 2 A 2 )=P(X 2 A 1 )P (Y 2 A 2 ) Useful to show that r.v. s are indep The joint density of (X, Y) can be written as: f(x, y) =h(x)k(y)

12 Ex 65 Example: two random variables that are independent y why? d The joint density of (X, Y) can be written as: c f(x, y) =h(x)k(y) a b x h(x) k(y) f(x, y) = 1 B(x, y) area(b) = 1[a,b] (x) area(b) 1 [c,d] (y)

13 Ex 66 Example: two random variables that are NOT independent why? y A2 For some intervals, A1, A2: P (X 2 A 1,Y 2 A 2 )=P (X 2 A 1 )P (Y 2 A 2 ) A1 x Pick A1, A2 as shown on the left Which one(s) of these are zero? (use material from earlier today) P (X 2 A 1,Y 2 A 2 ) P (X 2 A 1 )P )P (Y 2 A 2 )

14 Examples of nonuniform joint density

15 Ex 67a Example Suppose (X,Y) has joint density: f(x, y) =2e x 2y for x > 0 and y > 0 P(X > 1, Y < 1)?

16 Ex 67b Example Suppose (X,Y) has joint density: f(x, y) =2e x 2y for x > 0 and y > 0 P(X > 1, Y < 1)? P(X < Y)? X indep of Y? e -1 (1 - e -2 ) A. 1/2 B. 1/3 C. 1/4 D. 1/5

17 Ex 67b Example Suppose (X,Y) has joint density: f(x, y) =2e x 2y for x > 0 and y > 0 P(X > 1, Y < 1)? P(X < Y)? X indep of Y? e -1 (1 - e -2 ) A. 1/2 B. 1/3 C. 1/4 D. 1/5

18 A useful trick Known facts: Densities integrate to 1 For any λ > 0, λ exp(-λx) 1[0, )(x) is a density (the exponential density) Note: We can we use these two facts to get, without any effort: Z 1 0 e 5.2x dx = 1 5.2

19 Ex 53 Review: transformations Suppose I tell you that is the distribution of Richter scales What is the distribution of the amplitudes? For simplicity: Assume Richter scale X ~ Uniform(0, 1) What is the distribution of Y = exp(x)?

20 Review: recipe for transformations Suppose I tell you that is the distribution of Richter scales What is the distribution of the amplitudes? For simplicity: Assume Richter scale X ~ Uniform(0,1) What is the distribution of exp(x)? Recipe for finding the distribution of transforms of r.v. s 1 Find the CDF Density fx Richter: Amplitude: Differentiate to find the density

21 Review: recipe for transformations Suppose I tell you that is the distribution of Richter scales What is the distribution of the amplitudes? For simplicity: Assume Richter scale X ~ Uniform(0,1) What is the distribution of exp(x)? 1 Find the CDF F Y (y) =P (exp(x) apple y) = P (X apple log(y)) Why? = F X (log(y)) = 1 [1,e] (y) log(y)

22 Why P(exp(X) y) = P(X log(y)) Because (exp(x) y) = (X log(y)), which is true because: log is increasing, i.e. x 1 x 2 iff log(x 1 ) log(x 2 ) this means I can take log on both sides of the inequality: (exp(x) y) = (log(exp(x)) log(y)) log/exp are invertible: log(exp(z)) = z, so (log(exp(x)) log(y)) = (X log(y))

23 Review: recipe for transformations Suppose I tell you that is the distribution of Richter scales What is the distribution of the amplitudes? For simplicity: Assume Richter scale X ~ Uniform(0,1) What is the distribution of exp(x)? 2 Differentiate to find the density f Y (y) = df Y (y) dy at points where FY is differentiable =1 [1,e] (y) 1 y

24 Sums of independent discrete random variables (exact method)

25 Sum of independent r.v.s: summary Approximations: Central limit theorem (Normal approximation) Use software/ppl Exact methods: Binomial distribution (works only for sum of Bernoullis) Today: general, exact method CONVOLUTIONS

26 Ex 68 Simple example X: outcome of white dice Y: outcome of black dice Example: computing P(X + Y = 4)

27 Simple example

28 Application Not convinced? Play this game: Settler of Catan

29 Prop 16 General formula for discrete r.v.s If: Z = X + Y Then: p Z (z) = y = p X (z y) p Y (y).

30 Sums of independent continuous random variables

31 Sum of continuous r.v.s X: a continuous r.v. with density fx Y: a continuous r.v. with density fy Assume they are indep: f(x, y) = fx(x) fy(y) What is the density fz of the sum Z = X + Y? Recipe for finding the distribution of transforms of r.v. s 1 Find the CDF Density fx Richter: Amplitude: Differentiate to find the density

32 Ex 69 Example Let X and Y be independent and both uniform on [0, 1] y What is the density fz of the sum Z = X + Y? x

33 Example Let X and Y be independent and both uniform on [0, 1] y 1 Find the CDF What is the density fz of the sum Z = X + Y? x FZ(z) = P(Z z) example: z = 1 P( Z 1 ) = P( X + Y 1 ) =?

34 Example Let X and Y be independent and both uniform on [0, 1] y 1 Find the CDF P(Z z) for all z What is the density fz of the sum Z = X + Y? x example: z = 1 P( Z 1 ) = P( X + Y 1 ) y A = {(x,y) : x + y 1} = P( (X, Y) A ) = = Z A Z 1 = 1/2 f(x, y) dx dy Z x f(x, y) dy dx x

35 Example Let X and Y be independent and both uniform on [0, 1] y 1 Find the CDF P(Z z) for all z x What is the density fz of the sum Z = X + Y? P( Z z ) = P( X + Y z ) = Definition of the = CDF F(y) = Z 1 Z z 1 1 Z 1 1 Z 1 1 f X (x) x f X (x)f Y (y) dy Z z 1 f X (x)(f Y (z x f Y (y) dy x)) dx dx dx

36 Example Let X and Y be independent and both uniform on [0, 1] y x What is the density fz of the sum Z = X + Y? 1 Find the CDF Differentiate to find the 2 density FZ(z) = P( Z z ) f Z (z) = df Z(z) = dz Under regularity conditions, you = can interchange integrals and derivatives = = Z 1 1 Z 1 1 Z 1 1 Z 1 1 f X (x)f Y (z x)dx Chain rule of f X (x) df Y (z x) calculus dx dz d f X (x)f Y (z x) (z x) dx dz f X (x)f Y (z x)dx

37 Prop 16b Sum of continuous r.v.s X: a continuous r.v. with density fx Y: a continuous r.v. with density fy What is the density fz of the sum Z = X + Y? f Z (z) = = f X (z y) f Y (y) dy f X (x) f Y (z x) dx Terminology: convolution

38 Ex 69 Let X and Y be independent and both uniform on [0, 1] y What is the density fz of the sum Z = X + Y? x Note: Not equal to the sum of the densities!!!