CS  109Lecture  9

April  15th,  2016

Four Prototypical Trajectories

Review

Bernoulli:§ indicator  of  coin  flip  X  ~  Ber(p)

Binomial:  § #  successes  in  n coin  flips  X  ~  Bin(n,  p)

Poisson:  § #  successes  in  n coin  flips  X  ~  Poi(λ)

Geometric:  § #  coin  flips  until  success  X  ~  Geo(p)

Negative  Binomial:  § #  trials  until  r successes  X  ~  NegBin(r,  p)

Hyper  Geometric:  § #  white  balls  drawn  without  replacement  from  urn  with  N balls,  m  are  white:  X  ~  HypG(n,  N,  m)

Discrete Distributions

Supreme  Court  case:  Berghuis v.  SmithIf  a  group  is  underrepresented   in  a  jury  pool,  how  do  you  tell?

§ Article  by  Erin  Miller  – Friday,  January  22,  2010§ Thanks  to  (former  CS109er)  Josh  Falk  for  this  article

Justice  Breyer  [Stanford  Alum]  opened  the  questioning  by  invoking  the  binomial  theorem. He  hypothesized  a  scenario  involving  “an  urn  with  a  thousand  balls,  and  sixty  are  blue,  and  nine  hundred  forty  are  purple,  and  then  you  select  them  at  random…  twelve  at  a  time.” According  to  Justice  Breyer  and  the  binomial  theorem,  if  the  purple  balls  were  under  represented   jurors  then  “you  would  expect…  something   like  a  third  to  a  half  of  juries  would  have  at  least  one  minority  person” on  them.

Balls, Urns and the Supreme Court

• Should  model  this  combinatorially (X  ~  HypGeo)§ Ball  draws  not  independent   trials  (balls  not  replaced)

• Exact  solution:P(draw  12  purple  balls)  =                                              ≈ 0.4739

P(draw  ≥  1  blue  ball)  =  1  – P(draw  12  purple)  ≈ 0.5261

• Approximation  using  Binomial  distribution§ Assume  P(blue  ball)  constant  for  every  draw  =  60/1000§ X  =  #  blue  balls  drawn.    X  ~  Bin(12,  60/1000  =  0.06)§ P(X  ≥  1)  =  1  – P(X  =  0)  ≈ 1  – 0.4759  =  0.5240

In  Breyer’s  description,  should  actually  expect  just  over  half  of  juries  to  have  at  least  one  black  person  on  them

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

121000

12940

Justin Breyer Meets CS109

Demo

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0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 1 2 3 4 5 6 7 8 9 10 11 12

P(X

= x

)

# Underrepresented Jurrors

Underrepresented Juror PMF

• Determine  N =  how  many  of  some  species  remain§ Randomly  tag  m of  species  (e.g.,  with  white  paint)§ Allow  animals   to  mix  randomly  (assuming  no  breeding)§ Later,  randomly  observe  another  n of  the  species§ X  =  number  of  tagged  animals   in  observed  group  of  n§ X  ~  HypG(n,  N,  m)

• “Maximum  Likelihood”  estimate§ Set  N to  be  value  that  maximizes:

for  the  value  i of  X  that  you  observed  à =    mn/i• Calculated  by  assuming:  i =  E[X]  =  nm/N

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛

==

nN

inmN

im

iXP )(

Endangered Species

N̂

Four Prototypical Trajectories

End  Review

• 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§ 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

• Generally,  it  means  replace                        with  ∑=

b

axxf )( ∫

b

a

dxxf )(

From Discrete to Continuous

Integrals

*loving,  not  scary

• X is  a  Continuous  Random  Variable if  there  is  function f(x) ≥  0  for  -­∞ ≤ x ≤  ∞,  such  that:

• f is  a  Probability  Density  Function  (PDF)  if:

∫=≤≤b

adxxfbXaP )()(

1)()( ==∞<<−∞ ∫∞

∞−dxxfXP

Continuous Random Variables

• Say  f is  a  Probability  Density  Function (PDF)

§ f(x) is  not a  probability,   it  is  probability/units   of  X§ Not  meaningful  without  some  subinterval  over  X

§ Contrast  with  Probability  Mass  Function  (PMF)  in  discrete  case:

where for  X  taking  on  values  x1,  x2,  x3,  ...  

0)()( === ∫a

adxxfaXP

)()( aXPap ==

1)(1

=∑∞

=iixp

1)()( ==∞<<−∞ ∫∞

∞−dxxfXP

Probability Density Function

• For  a  continuous  random  variable  X,  the  Cumulative  Distribution  Function (CDF)  is:

• Density  f is  derivative  of  CDF  F:

• For  continuous f and  small  ε :

§ So,  ratio of  probabilities   can  still  be  meaningful:o P(X  =  1)/P(X  =  2)  ≈ (ε f(1))/(ε f(2))  = f(1)/f(2)

∫+

−

≈=+≤≤−2/

2/

)()()( 22

ε

ε

εεεa

a

afdxxfaXaP

∫∞−

=≤=<=a

dxxfaXPaXPaF )( )()()(

)()( aFdadaf =

Cumulative Distribution Function

• X  is  continuous  random  variable  (CRV)  with  PDF:

§ What  is  C?

§ What  is  P(X  >  1)?

⎩⎨⎧ <<−

= otherwise 0

2 x 0n whe)24()(

2xxCxf

13

22 1)24(0

232

2

0

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛−⇒=−∫xxCdxxxC

83 1

38 10

3168 =⇒=⇒=⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛ − CCC

21

322

3168

83

322

83)24()(

1

232

2

1

2

183 =⎥

⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −−⎟⎠

⎞⎜⎝

⎛ −=⎟⎟⎠

⎞⎜⎜⎝

⎛−=−= ∫∫

∞ xxdxxxdxxf

Simple Example

• X  =  days  of  use  before  your  disk  crashes

§ First,  determine  λ to  have  actual  PDFo Good  integral  to  know:

§ What  is  P(50  <  X  <  150)?

§ What  is  P(X  <  10)?

⎩⎨⎧ ≥

=−

otherwise0 0

)(100/ xe

xfxλ

uu edue =∫100

1100

1 10010010010

100/100/100/ =⇒=−=−==∞−−− ∫∫ − λλλλλ xxx edxedxe

383.0 )50()150( 2/12/3

50

150100/150

50

100/100

1 ≈+−=−==− −−−−∫ eeedxeFF xx

095.01 )10( 10/1

0

10100/10

0

100/100

1 ≈+−=−== −−−∫ eedxeF xx

Disk Crashes

For  continuousRV  X:For  discrete RV  X:

∑=x

xpxXE )( ][

222 ])[(][])[()(Var XEXEXEX −=−= µ

dxxfxXE ∫∞

∞−

= )( ][

dxxfxgXgE ∫∞

∞−

= )() ()]([∑=x

xpxgXgE )( )()]([

bXaEbaXE +=+ ][][

∑=x

nn xpxXE )( ][ dxxfxXE nn ∫∞

∞−

= )( ][

)(Var)(Var 2 XabaX =+

For  both discrete  and  continuous  RVs:

Expectation and Variance

• X  is  a  continuous  random  variable  with  PDF:

§ What  is  E[X]?

§ What  is  Var(X)?  

⎩⎨⎧ ≤≤

=otherwise0

10 2)( xxxf

32

322)(][

013

1

0

2 ==== ∫∫

∞

∞−

xdxxdxxfxXE

x

)(xf

21

212)(][

014

1

0

322 ==== ∫∫

∞

∞−

xdxxdxxfxXE

181

32

21])[(][)(

222 =⎟

⎠

⎞⎜⎝

⎛−=−= XEXEXVar

Linearly Increasing Density

• X  is  a  Uniform  Random  Variable:  X  ~  Uni(α,  β)§ Probability  Density  Function  (PDF):

o Sometimes  defined  over  range  α <  x <  β

§ (for  α ≤  a ≤ b ≤  β)

§

§

⎪⎩

⎪⎨⎧ ≤≤

= −

otherwise0

)(1

βααβ xxf

αβ −−

==≤≤ ∫abdxxfbxaP

b

a

)()(

2)(2)(2)(][

222

βααβαβ

αβ α

ββ

ααβ

+=

−

−=

−=== ∫∫ −

∞

∞−

xdxdxxfxXE x

12)()(2αβ −

=XVar

αβ −1

x

)(xf

Uniform Random Variable

• X  ~  Uni(0,  20)

§ P(X  <  6)?

§ P(4  <  X  <  17)?

206

201)6(

6

0

==< ∫ dxxP

2013

204

2017

201)174(

17

4

=−==<< ∫ dxxP

⎪⎩

⎪⎨⎧ ≤≤

=otherwise0

200 )( 201

xxf

Fun with the Uniform Distribution

Riding the Marguerite

• Say  the  Marguerite  bus  stops  at  the  Gates  bldg.  at  15  minute  intervals  (2:00,  2:15,  2:30,  etc.)§ Passenger  arrives  at  stop  uniformly  between  2-­2:30pm§ X  ~  Uni(0,  30)

• P(Passenger  waits  <  5  minutes  for  bus)?§ Must  arrive  between  2:10-­2:15pm  or  2:25-­2:30pm

• P(Passenger  waits  >  14  minutes  for  bus)?§ Must  arrive  between  2:00-­2:01pm  or  2:15-­2:16pm

31

305

305)3025()1510(

30

25301

15

10301 =+=+=<<+<< ∫∫ dxdxxPXP

151

301

301)1615()10(

16

15301

1

0301 =+=+=<<+<< ∫∫ dxdxxPXP

Riding the Marguerite

• Biking  to  a  class  on  campus§ Leave  t minutes  before  class  starts§ X  =  travel  time  (minutes). X  has  PDF:    f(x)§ If  early,  incur  cost:  c/min. If  late,  incur  cost:  k/min.

§ Choose   t (when  to  leave)  to  minimize  E[C(X,  t)]:

dxxftxkdxxfxtcdxxftXCtXCEt

t

)( )()( )()( ),()],([00

∫∫∫∞∞

−+−==

⎪⎩

⎪⎨⎧

≥−

<−=

txtXktxXtc

tXC if)(

if )(),( :Cost

When to Leave for Class

Minimization via Differentiation• Want  to  minimize  w.r.t.  t:

§ Differentiate  E[C(X,  t)]  w.r.t.  t,  and  set  =  0  (to  obtain  t*):o Leibniz  integral  rule:

∫∫ ∂

∂+−=

)(

)(1

12

2)(

)(

2

1

2

1

),()),(()()),(()(),(tf

tf

tf

tf

dxttxgttfg

dttdfttfg

dttdfdxtxg

dtd

∫∫∞

−−−+−=t

t

dxxkftfttkdxxcftfttctXCEdtd )()()()()()()],([

0

dxxftxkdxxfxtctXCEt

t

)( )()( )()],([0

∫∫∞

−+−=

kcktFtFktcF+

=⇒−−= *)( *)](1[*)(0