CS145: Probability & Computing Lecture 11: Derived Distributions, Functions of Random Variables

Transcription

1 CS145: Probability & Computing Lecture 11: Derived Distributions, Functions of Random Variables Instructor: Erik Sudderth Brown University Computer Science March 5, 2015

2 Homework Submissions Electronic Submission Rules: You must use the handin program to submit any code you write, plus a single pdf with all other plots, calculations, and answers. Late Submissions: If you don t handin a properly formatted submission by Thursday at 11:59pm, it is late. We will strictly enforce this on all future assignments. File modification times are not proof of on-time completion. Late Submission penalties: No penalty for first two late submissions. Can submit more late homeworks with a 20-point penalty (corresponds to 1% of overall course score). Zero credit after Monday at midnight (>4 days late).

3 Midterm Exam Thursday, March 12 from 2:30-3:50pm in Metcalf Auditorium (room 101). Material: Up to and including the lecture today (March 5). Bertsekas & Tsitsiklis Chapters 1, 2, 3, 4.1, 4.3. Format: Hand-written answers to probability problems. Similar to medium difficulty problems from homeworks. No programming, computers not needed or allowed. Formula Sheet: We will distribute one early next week. Included: Formulas for the pmf/pdf/cdf of particular distributions, integral and derivative identities, etc. Not included: Conceptual equations. Joint and marginal distributions, Bayes rule, computing cdf from pdf, etc.

4 Derived Distributions Some figures and materials courtesy Bertsekas & Tsitsiklis, Introduction to Probability, 2008

5 Functions of Random Variables Suppose we care about some function of random variable X: Y = g(x) We know how to find the expected value: Continuous: Z E[Y ]=E[g(X)] = We can also find the distribution of Y as follows: Find the CDF of Y: x g(x)f X (x) dx F Y (y) =P (Y apple y) =P (g(x) apple y) If desired, differentiate to find the PDF of Y: f Y (y) = df Y (y) dy

6 ( ) Example: Inverse of a Uniform ( Variable ) Your friend is going to take the train 180 miles from Providence to New York. In bad weather, their speed X is uniformly distributed between 30 and 60 mph. What is the probability density of their travel time Y? { 1 30 PDF f X (x) 1 CDF F X (x) F Y (y) = { 0 if y 3, 2 (6/y) if 3 y 6, 1 if 6 y, PDF f Y (y) x Y = 180 X CDF F Y (y) y 3 6 y 1 x f Y (y) = { 0 if y 3, { 6/y 2 if 3 y 6, 0 if 6 y. E[Y ] = 6 log(2) 4.16 > 180 E[X]

7 Distributions of Linear Functions Y =2X + 5: f X f ax fax+b If Y = ax + b, a 6= 0, then 1 y b f Y (y) = f a X a y If a>0,f Y (y) =F X a b If a<0,f Y (y) =1 F X y b a

8 Scaling a Uniform Variable 2 Density of X Density of 2X Density of 2X o o o 1 2 Density of 0.5X Density of 0.5X + 1 Density of -0.5X o 2-1 o o 1 2 Normal distributions.

9 Scaling an Exponential Variable We often model waiting or arrival times X as: f X (x) = e E[X] = 1 x,x 0. Consider a waiting time with scaled units: Y = ax, a > 0. f Y (y) = 1 a f X y a = a e ( /a)x E[Y ]= a This random variable is still exponentially distributed! What about a general linear function? Y = ax + b

10 f X (x) = Scaling a Gaussian Variable 1 p 2 2 e 1 2( x µ ) 2 f X (x) E[X] =µ Var[X] =E[(X µ) 2 ]= 2 p Var[X] = is the standard deviation Any linear transformation of a Gaussian variable is Gaussian! 1 p e 1 2( y µ ) Y = ax + b f Y (y) = µ = aµ + b, = a

11 Standard Normal Random Variables If X N(µ, 2 ) then for any constants a and b the random variable ax + b is distributed N(aµ + b, a 2 2 ). If X N(µ, 2 ) then Z = X µ is distribution N(0, 1) N(0, 1) is the standard Normal distribution. Pr(Z apple z) = Z (z) = 1 p 2 Z z 1 e t2 /2 dt We can use this to evaluate normal CDF: Pr(X apple x) =Pr( X µ apple x µ )= ( x µ ) ( x µ x µ )=1 ( ) Matlab uses this same transformation: Probability Content from -oo to Z Z

12 / Invertible & Smooth Transformations Suppose Y=g(X) is monotonic ( ) and thus invertible: y = g(x) ( if and only) if x ( = h(y) ) If it is also differentiable, we have: ( ) f Y (y) =f X h(y) dh dy (y) y Event {X < h(y)} To prove for a monotonically increasing function: F Y (y) =P ( g(x) y ) = P ( X h(y) ) ( ) = F X h(y) f Y (y) = df Y dy (y) =f X ( h(y) )dh dy (y). y= g(x) h(y ) x y h(y ) ( ) y= g(x) x Event {X >h(y)}

13 Example: Square Root of Exponential X has an exponential distribution: Y = p X f X (x) =e x,x 0. f Y (y) =2ye y2,y 0. This is a Rayleigh distribution. It arises in many applications where we observation is the magnitude of some vector (wind speed, MRI, etc.) --Y---I Y = g(x) y+ dy= g(x+ dx) x+ dx 4

14 Non-invertible Transformations You strike a pool ball with velocity X uniformly distributed between -1 and 1. What is the distribution of the ball s kinetic energy Y? Y = X 2 f Y (y) = 1 2 p y [f X( p y)+f X ( f Y (y) = 1 2 p y p y)] Energy = 1 hat x = y, Mass Velocity2 2 = 2x, General functions: y x=+fii y = g(x) {x:g(xt = y} /