m 2 s 2 2 ( ( x ) / 2 x / 2 ) ( ) - - PDF document

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m 2 s 2 2 ( ( x ) / 2 x / 2 ) ( ) - - PDF document

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Normal Random Variable Carl Friedrich Gauss X is a Normal Random Variable : X ~ N( m , s 2 ) Carl Friedrich Gauss (1777-1855) was a remarkably influential German mathematician Probability Density Function (PDF): 1 2 2

slide-1
SLIDE 1

1 Normal Random Variable

  • X is a Normal Random Variable: X ~ N(m, s 2)
  • Probability Density Function (PDF):
  • Also called “Gaussian”
  • Note: f(x) is symmetric about m
  • Common for natural phenomena: heights, weights, etc.
  • Often results from the sum of multiple variables

     

 

x e x f

x

where 2 1 ) (

2 2 2

/ ) ( s m

 s m  ] [X E

2

) ( s  X Var

) (x f x m

Carl Friedrich Gauss

  • Carl Friedrich Gauss (1777-1855) was a

remarkably influential German mathematician

  • Started doing groundbreaking math as teenager
  • Did not invent Normal distribution, but popularized it
  • He looked more like Martin Sheen
  • Who is, of course, Charlie Sheen’s father

Properties of Normal Random Variable

  • Let X ~ N(m, s 2)
  • Let Y = aX + b
  • Y ~ N(am + b, a2s 2)
  • E[Y] = E[aX + b] = aE[X] + b = am + b
  • Var(Y) = Var(aX + b) = a2Var(X) = a2s 2

Differentiating FY(x) w.r.t. x , yields fY(x), the PDF for y:

  • Special case: Z = (X – m)/s

(a = 1/s , b = –m/s)

  • Z ~ N(am + b, a2s 2) = N(m/s – m/s, (1/s )2s2) = N(0, 1)

) ( ) ( ) ( ) ( ) (

a b x a b x

X Y

F X P x b aX P x Y P x F

 

        ) ( ) ( ) ( ) (

1

a b x a a b x dx d dx d

X X Y Y

f F x F x f

 

  

Standard (Unit) Normal Random Variable

  • Z is a Standard (or Unit) Normal RV: Z ~ N(0, 1)
  • E[Z] = m = 0

Var(Z) = s 2 = 1 SD(Z) = s = 1

  • CDF of Z, FZ(z) does not have closed form
  • We denote FZ(z) as (z): “phi of z”
  • By symmetry: (–z) = P(Z ≤ –z) = P(Z ≥ z) = 1 – (z)
  • Use Z to compute X ~ N(m, s 2), where s > 0
  • Table of (z) values in textbook, p. 201 and handout

dx e dx e z Z P z

z x z x

 

      

   

2 / 2 / ) (

2 2 2

2 1 2 1 ) ( ) Φ(   s

s m

) ( ) ( ) ( ) ( ) (

s s s s

m m m m    

       

x x x X

Z P P x X P x FX

Using Table of (z) Values

(0.54) = 0.7054

  • X ~ N(3, 16)

m = 3 s 2 = 16 s = 4

  • What is P(X > 0)?
  • What is P(2 < X < 5)?
  • What is P(|X – 3| > 6)?

Get Your Gaussian On

) 4 2 4 1 ) 4 3 5 4 3 4 3 2

( ( ) 5 2 (        

   

Z P P X P

X

2902 . ) 5987 . 1 ( 6915 . )) ( 1 ( ) ( ) ( ) (

4 1 2 1 4 1 4 2

             7734 . ) ( ) ( 1

4 3 4 3

     

) 4 3 ) 4 3 4 3

( ( ) (    

   

Z P P X P

X ) 4 3 9 ) 4 3 3

( ( ) 9 ( ) 3 (

    

      Z P Z P X P X P 1336 . ) 9332 . 1 ( 2 )) ( 1 ( 2 )) ( 1 ( ) (

2 3 2 3 2 3

          

slide-2
SLIDE 2

2

  • Send voltage of 2 or -2 on wire (to denote 1 or 0)
  • X = voltage sent
  • R = voltage received = X + Y, where noise Y ~ N(0, 1)
  • Decode R: if (R ≥ 0.5) then 1, else 0
  • What is P(error after decoding | original bit = 1)?
  • What is P(error after decoding | original bit = 0)?

Noisy Wires

0668 . ) 5 . 1 ( 1 ) 5 . 1 ( ) 5 . 1 ( ) 5 . 2 (             Y P Y P 0062 . ) 5 . 2 ( 1 ) 5 . 2 ( ) 5 . 2 (          Y P Y P

  • X ~ Bin(n, p)
  • E[X] = np

Var(X) = np(1 – p)

  • Poisson approx. good: n large (> 20), p small (< 0.05)
  • For large n: X  Y ~ N(E[X], Var(X)) = N(np, np(1 – p))
  • Normal approx. good : Var(X) = np(1 – p) ≥ 10
  • DeMoivre-Laplace Limit Theorem:
  • Sn: number of successes (with prob. p) in n independent trials

Normal Approximation to Binomial

 )

                                ) 1 ( 5 . ) 1 ( 5 . ) (

2 1 2 1

p np np k p np np k k Y k P k X P ) ( ) ( ) 1 ( a b b p np np S a P

n n

                  

 

Comparison when n = 100, p = 0.5

P(X = k) k

  • 100 people placed on special diet
  • X = # people on diet whose cholesterol decreases
  • Doctor will endorse diet if X ≥ 65
  • What is P(doctor endorses diet | diet has no effect)?
  • X ~ Bin(100, 0.5)
  • Use Normal approximation: Y ~ N(50, 25)
  • Using Binomial:

Faulty Endorsements

) 5 . 64 ( ) 65 (    Y P X P

 )

0019 . ) 9 . 2 ( 1 ) 5 . 64 (

5 50 5 . 64 5 50

      

  Y

P Y P 0018 . ) 65 (   X P 5 1 25 1 50 – p) np( – p) np( np   

  • Stanford accepts 2480 students
  • Each accepted student has 68% chance of attending
  • X = # students who will attend. X ~ Bin(2480, 0.68)
  • What is P(X > 1745)?
  • Use Normal approximation: Y ~ N(1686.4, 539.65)
  • Using Binomial:

Stanford Admissions

) 5 . 1745 ( ) 1745 (    Y P X P 23 . 23 1 65 . 539 1 4 . 1686 – p) np( – p) np( np   

 )

0055 . ) 54 . 2 ( 1 ) 5 . 1745 (

23 . 23 4 . 1686 5 . 1745 23 . 23 4 . 1686

      

  Y

P Y P 0053 . ) 1745 (   X P

Exponential Random Variable

  • X is an Exponential RV: X ~ Exp(l) Rate: l > 0
  • Probability Density Function (PDF):
  • Cumulative distribution function (CDF), F(X) = P(X  x):
  • Represents time until some event
  • Earthquake, request to web server, end cell phone contract, etc.

          

x where if if ) ( x x e x f

x l

l l 1 ] [  X E

2

1 ) ( l  X Var

) (x f x

where 1 ) (   

x e x F

x l

slide-3
SLIDE 3

3

  • X = time until some event occurs
  • X ~ Exp(l)
  • What is P(X > s + t | X > s)?
  • After initial period of time s, P(X > t | ) for waiting

another t units of time until event is same as at start

  • “Memoryless” = no impact from preceding period s

Exponential is “Memoryless”

) ( ) ( ) ( ) and ( ) | ( s X P t s X P s X P s X t s X P s X t s X P             ) ( ) ( 1 ) ( 1 ) ( 1 ) ( ) (

) (

t X P t F e e e s F t s F s X P t s X P

t s t s

            

    l l l

) ( ) | ( So, t X P s X t s X P     

  • Say a visitor to your web leaves after X minutes
  • On average, visitors leave site after 5 minutes
  • Assume length of stay is Exponentially distributed
  • X ~ Exp(l = 1/5), since E[X] = 1/l = 5
  • What is P(X > 10)?
  • What is P(10 < X < 20)?

Visits to Web Site

1353 . ) 1 ( 1 ) 10 ( 1 ) 10 (

2 10

       

 

e e F X P

l

1170 . ) 1 ( ) 1 ( ) 10 ( ) 20 ( ) 20 10 (

2 4

        

 

e e F F X P

  • X = # hours of use until your laptop dies
  • On average, laptops die after 5000 hours of use
  • X ~ Exp(l = 1/5000), since E[X] = 1/l = 5000
  • You use your laptop 5 hours/day.
  • What is P(your laptop lasts 4 years)?
  • That is: P(X > (5)(365)(4) = 7300)
  • Better plan ahead... especially if you are coterming:

Replacing Your Laptop

2322 . ) 1 ( 1 ) 7300 ( 1 ) 7300 (

46 . 1 5000 / 7300

       

 

e e F X P plan) year (5 1612 . ) 9125 ( 1 ) 9125 (

825 . 1

    

e F X P plan) year (6 1119 . ) 10950 ( 1 ) 10950 (

19 . 2

    

e F X P

  • Product rule for derivatives:
  • Derivative and integral of exponential:
  • Integration by parts:

A Little Calculus Review

dv u v du v u d      ) (

dx du e dx e d

u u

 ) (

  

       dv u du v v u v u d ) (

 

     du v v u dv u

u u

e du e 

  • Compute n-th moment of Exponential distribution
  • Step 1: don’t panic, think happy thoughts, recall...
  • Step 2: find u and v (and du and dv):
  • Step 3: substitute (a.k.a. “plug and chug”)

And Now, Some Calculus Practice

 

 ] [ dx e x X E

x n n l

l

x n

e v x u

l 

   dx e dv dx nx du

x n l

l 

 

1

   

   

          dx e nx e x du v v u dx e x dv u

x n x n x n l l l

l

1

] [ ] [

1 1 1      

     

 

n x n x n x n n

X E dx e x dx e nx e x X E

n n l l

l l l

l ,... ] [ ] [ so , 1 ] 1 [ ] [ : case Base

2 2

2 1 2 , 1 l l l l

     X E X E E X E