Demo Have (uncountably) infinite values (e.g., real numbers) - PDF document

Balls, Urns, and the Supreme Court Justice Breyer Meets CS109 • Supreme Court case: Berghuis v. Smith • Should model this combinatorially If a group is underrepresented in a jury pool, how do you tell?  Ball draws not independent trials (balls not replaced)  Article by Erin Miller – Friday, January 22, 2010 • Exact solution:     940 1000      Thanks to Josh Falk for pointing out this article P(draw 12 black balls) =  0.4739         12 12 Justice Breyer [Stanford Alum] opened the questioning by P(draw ≥ 1 red ball) = 1 – P(draw 12 black balls)  0.5261 invoking the binomial theorem. He hypothesized a scenario involving “an urn with a thousand balls, and sixty are red, • Approximation using Binomial distribution and nine hundred forty are black, and then you select them  Assume P(red ball) constant for every draw = 60/1000 at random… twelve at a time.” According to Justice Breyer  X = # red balls drawn. X ~ Bin(12, 60/1000 = 0.06) and the binomial theorem, if the red balls were black jurors then “you would expect… something like a third to a half of  P(X ≥ 1 ) = 1 – P(X = 0)  1 – 0.4759 = 0.5240 juries would have at least one black person” on them. In Breyer’s description, should actually expect just over half Justice Scalia’s rejoinder: “We don’t have any urns here.” • of juries to have at least one black person on them From Discrete to Continuous • So far, all random variables we saw were discrete  Have finite or countably infinite values (e.g., integers)  Usually, values are binary or represent a count • Now it’s time for continuous random variables Demo  Have (uncountably) infinite values (e.g., real numbers)  Usually represent measurements (arbitrary precision) o Height (centimeters), Weight (lbs.), Time (seconds), etc. • Difference between how many and how much b b   • Generally, it means replace with f ( x ) f ( x ) dx  x a a Continuous Random Variables Probability Density Functions • X is a Continuous Random Variable if there is • Say f is a Probability Density Function (PDF) function f ( x ) ≥ 0 for -  ≤ x ≤  , such that:         P ( X ) f ( x ) dx 1    b    P ( a X b ) f ( x ) dx  f ( x ) is not a probability, it is probability/units of X a  Not meaningful without some subinterval over X  a • f is a Probability Density Function (PDF) if:    P ( X a ) f ( x ) dx 0  a        P ( X ) f ( x ) dx 1    Contrast with Probability Mass Function (PMF) in   discrete case: p ( a ) P ( X a )    where p ( x ) 1 for X taking on values x 1 , x 2 , x 3 , ... i  i 1 1

Cumulative Distribution Functions Simple Example • For a continuous random variable X, the • X is continuous random variable (CRV) with PDF: Cumulative Distribution Function (CDF) is:     2 C ( 4 x 2 x ) whe n 0 x 2   f ( x ) a   0 otherwise      F ( a ) P ( X a ) P ( X a ) f ( x ) dx  What is C ?   d   2 3  2 x 2    • Density f is derivative of CDF F : ( ) ( )      f a F a 2 2 C ( 4 x 2 x ) dx 1 C  2 x  1   da 3 0 0      16 8 3 • For continuous f and small  :           8 0 1 1 C   C C    3  3 8   a / 2           P ( a X a ) f ( x ) dx f ( a )  What is P(X > 1)? 2 2              a / 2   2 3 3 2 x 2 3 16 2 1  3         2 2     f ( x ) dx ( 4 x 2 x ) dx  2 x   8 2  8         8 3 8 3 3 2 1 1 1 Disk Crashes Expectation and Variance • X = hours before your disk crashes For discrete RV X : For continuous RV X :        x / 100 e x 0    E [ X ] x f ( x ) dx  E [ X ] x p ( x ) f ( x )  0 otherwise   x   First, determine  to have actual PDF     E [ g ( X )] g ( x ) p ( x ) E [ g ( X )] g ( x ) f ( x ) dx   u u o Good integral to know: e du e x             1           1  x / 100 x / 100 x / 100 1 e dx 100 e dx 100 e 100     100 100 n n E [ X ] x p ( x ) n n 0 E [ X ] x f ( x ) dx  What is P(50 < X < 150)? x   150 For both discrete and continuous RVs:  1         150     x / 100 x / 100 3 / 2 1 / 2 F ( 150 ) F ( 50 ) e dx e e e 0 . 383 100    50 E [ aX b ] aE [ X ] b 50  What is P(X < 10)?      2 2 2 Var ( X ) E [( X ) ] E [ X ] ( E [ X ]) 10   1     10      x / 100 x / 100 1 / 10 F ( 10 ) e dx e e 1 0 . 095 100   0 2 Var ( aX b ) a Var ( X ) 0 Linearly Increasing Density Uniform Random Variable • X is a Uniform Random Variable : X ~ Uni( a , b ) • X is a continuous random variable with PDF:  Probability Density Function (PDF):  2 x   0 x 1  f ( x )  f ( x )  1 f ( x )  a   b  0 otherwise b  a x   f ( x ) 1  b  x  a 0 otherwise  What is E[X]? x  b  1 b a 2 2            2 3  P ( a x b ) f ( x ) dx [ ] ( ) 2 1 E X x f x dx x dx x b  a 3 0 3   a 0   b  a a  b 2 2  What is Var(X)?     x    E [ X ] x f ( x ) dx dx  1 b  a b  a 1 1   2 ( ) 2     2 2 3 4 1     E [ X ] x f ( x ) dx 2 x dx x 2 0 2 b  a   2 0 ( )   Var ( X ) 2   1 2 1 12        2 2 Var ( X ) E [ X ] ( E [ X ])   2 3 18 2

Fun with the Uniform Distribution Riding the Marguerite Bus • X ~ Uni(0, 20) • Say the Marguerite bus stops at the Gates bldg. at 15 minute intervals (2:00, 2:15, 2:30, etc.)  1    0 x 20  20  f ( x )  Passenger arrives at stop uniformly between 2-2:30pm   0 otherwise  X ~ Uni(0, 30)  P(X < 6)? • P(Passenger waits < 5 minutes for bus)? 6 1 6     P ( x 6 ) dx  Must arrive between 2:10-2:15pm or 2:25-2:30pm 20 20 15 30 5 5 1 0             1 1 P ( 10 X 15 ) P ( 25 x 30 ) dx dx 30 30 30 30 3  P(4 < X < 17)? 10 25 17 • P(Passenger waits > 14 minutes for bus)? 1 17 4 13        P ( 4 x 17 ) dx 20 20 20 20  Must arrive between 2:00-2:01pm or 2:15-2:16pm 4 1 16 1 1 1             1 1 P ( 0 X 1 ) P ( 15 x 16 ) dx dx 30 30 30 30 15 0 15 When to Leave For Class Minimization via Differentiation • Biking to a class on campus • What to minimize w.r.t. t :  t  Leave t minutes before class starts       E [ C ( X , t )] c ( t x ) f ( x ) dx k ( x t ) f ( x ) dx  X = travel time (minutes). X has PDF: f ( x ) 0 t  If early, incur cost: c/min. If late, incur cost: k/min.  Differentiate E[C( X , t )] w.r.t. t , and set = 0 (to obtain t* ):    c ( t X )  o Leibniz integral rule: if x t   Cost : C ( X , t )    f ( t ) f ( t )   k ( X t ) if x t d 2 df ( t ) df ( t ) 2 g ( x , t )      2 1 g ( x , t ) dx g ( f ( t ), t ) g ( f ( t ), t ) dx 2 1   Choose t (when to leave) to minimize E[C( X , t )]: dt dt dt t f ( t ) f ( t )   1 1 t         [ ( , )] ( , ) ( ) ( ) ( ) ( ) ( )  E C X t C X t f x dx c t x f x dx k x t f x dx t d         E [ C ( X , t )] c ( t t ) f ( t ) cf ( x ) dx k ( t t ) f ( t ) kf ( x ) dx 0 0 t dt 0 t k      0 cF ( t *) k [ 1 F ( t *)] F ( t *)  c k 3