CS70: Jean Walrand: Lecture 34. Uniformly at Random in [ 0 , 1 ] . - PowerPoint PPT Presentation

CS70: Jean Walrand: Lecture 34. Uniformly at Random in [ 0 , 1 ] . Uniformly at Random in [ 0 , 1 ] . Let [ a , b ] denote the event that the point X is in the interval [ a , b ] . Choose a real number X , uniformly at random in [ 0 , 1 ] . Pr [[ a , b ]] = length of [ a , b ] length of [ 0 , 1 ] = b − a What is the probability that X is exactly equal to 1 / 3? Well, ..., 0. = b − a . 1 Intervals like [ a , b ] ⊆ Ω = [ 0 , 1 ] are events. Continuous Probability 1 More generally, events in this space are unions of intervals. Example: the event A - “within 0 . 2 of 0 or 1” is A = [ 0 , 0 . 2 ] ∪ [ 0 . 8 , 1 ] . What is the probability that X is exactly equal to 0 . 6? Again, 0. Thus, 1. Examples In fact, for any x ∈ [ 0 , 1 ] , one has Pr [ X = x ] = 0. 2. Events Pr [ A ] = Pr [[ 0 , 0 . 2 ]]+ Pr [[ 0 . 8 , 1 ]] = 0 . 4 . How should we then describe ‘choosing uniformly at random in [ 0 , 1 ] ’? 3. Continuous Random Variables Here is the way to do it: More generally, if A n are pairwise disjoint intervals in [ 0 , 1 ] , then Pr [ X ∈ [ a , b ]] = b − a , ∀ 0 ≤ a ≤ b ≤ 1 . Pr [ ∪ n A n ] := ∑ Pr [ A n ] . n Makes sense: b − a is the fraction of [ 0 , 1 ] that [ a , b ] covers. Many subsets of [ 0 , 1 ] are of this form. Thus, the probability of those sets is well defined. We call such sets events. Uniformly at Random in [ 0 , 1 ] . Uniformly at Random in [ 0 , 1 ] . Uniformly at Random in [ 0 , 1 ] . Note: Pr [ X ≤ x ] = x for x ∈ [ 0 , 1 ] . Also, Pr [ X ≤ x ] = 0 for x < 0 and Pr [ X ≤ x ] = 1 for x > 1. Let us define F ( x ) = Pr [ X ≤ x ] . Note: A radical change in approach. For a finite probability space, Ω = { 1 , 2 ,..., N } , we started with Pr [ ω ] = p ω . We then defined Pr [ X ∈ ( a , b ]] = Pr [ X ≤ b ] − Pr [ X ≤ a ] = F ( b ) − F ( a ) . Pr [ A ] = ∑ ω ∈ A p ω for A ⊂ Ω . We used the same approach for countable Ω . An alternative view is to define f ( x ) = d dx F ( x ) = 1 { x ∈ [ 0 , 1 ] } . Then For a continuous space, e.g., Ω = [ 0 , 1 ] , we cannot start with Pr [ ω ] , � b because this will typically be 0. Instead, we start with Pr [ A ] for some F ( b ) − F ( a ) = a f ( x ) dx . events A . Here, we started with A = interval, or union of intervals. Then we have Pr [ X ∈ ( a , b ]] = Pr [ X ≤ b ] − Pr [ X ≤ a ] = F ( b ) − F ( a ) . Thus, the probability of an event is the integral of f ( x ) over the event: Thus, F ( · ) specifies the probability of all the events! � Pr [ X ∈ A ] = A f ( x ) dx .

Uniformly at Random in [ 0 , 1 ] . Uniformly at Random in [ 0 , 1 ] . Nonuniformly at Random in [ 0 , 1 ] . Discrete Approximation: Fix N ≫ 1 and let ε = 1 / N . Think of f ( x ) as describing how Define Y = n ε if ( n − 1 ) ε < X ≤ n ε for n = 1 ,..., N . � ∞ one unit of probability is spread over [ 0 , 1 ] : uniformly! This figure shows a different choice of f ( x ) ≥ 0 with − ∞ f ( x ) dx = 1. Then | X − Y | ≤ ε and Y is discrete: Y ∈ { ε , 2 ε ,..., N ε } . Then Pr [ X ∈ A ] is the probability mass over A . It defines another way of choosing X at random in [ 0 , 1 ] . Also, Pr [ Y = n ε ] = 1 N for n = 1 ,..., N . Observe: Note that X is more likely to be closer to 1 than to 0. Thus, X is ‘almost discrete.’ � x − ∞ f ( u ) du = x 2 for x ∈ [ 0 , 1 ] . ◮ This makes the probability automatically additive. One has Pr [ X ≤ x ] = � x + ε � ∞ Also, Pr [ X ∈ ( x , x + ε )] = f ( u ) du ≈ f ( x ) ε . ◮ We need f ( x ) ≥ 0 and − ∞ f ( x ) dx = 1. x Another Nonuniform Choice at Random in [ 0 , 1 ] . General Random Choice in ℜ Pr [ X ∈ ( x , x + ε )] Let F ( x ) be a nondecreasing function with F ( − ∞ ) = 0 and F (+ ∞ ) = 1. An illustration of Pr [ X ∈ ( x , x + ε )] ≈ f X ( x ) ε : Define X by Pr [ X ∈ ( a , b ]] = F ( b ) − F ( a ) for a < b . Also, for a 1 < b 1 < a 2 < b 2 < ··· < b n , Pr [ X ∈ ( a 1 , b 1 ] ∪ ( a 2 , b 2 ] ∪ ( a n , b n ]] = Pr [ X ∈ ( a 1 , b 1 ]]+ ··· + Pr [ X ∈ ( a n , b n ]] = F ( b 1 ) − F ( a 1 )+ ··· + F ( b n ) − F ( a n ) . Let f ( x ) = d dx F ( x ) . Then, This figure shows yet a different choice of f ( x ) ≥ 0 with � ∞ − ∞ f ( x ) dx = 1. Pr [ X ∈ ( x , x + ε ]] = F ( x + ε ) − F ( x ) ≈ f ( x ) ε . It defines another way of choosing X at random in [ 0 , 1 ] . Note that X is more likely to be closer to 1 / 2 than to 0 or 1. Here, F ( x ) is called the cumulative distribution function (cdf) of X and � 1 / 3 x 2 � 1 / 3 = 2 For instance, Pr [ X ∈ [ 0 , 1 / 3 ]] = 4 xdx = 2 � 9 . f ( x ) is the probability density function (pdf) of X . 0 0 Thus, the pdf is the ‘local probability by unit length.’ Thus, Pr [ X ∈ [ 0 , 1 / 3 ]] = Pr [ X ∈ [ 2 / 3 , 1 ]] = 2 To indicate that F and f correspond to the RV X , we will write them 9 and It is the ‘probability density.’ F X ( x ) and f X ( x ) . Pr [ X ∈ [ 1 / 3 , 2 / 3 ]] = 5 9 .

Discrete Approximation Example: CDF Calculation of event with dartboard.. Example: hitting random location on gas tank. Random location on circle. Probability between . 5 and . 6 of center? 1 Recall CDF. Fix ε ≪ 1 and let Y = n ε if X ∈ ( n ε , ( n + 1 ) ε ] . y Thus, Pr [ Y = n ε ] = F X (( n + 1 ) ε ) − F X ( n ε ) .  0 for y < 0  y 2 Note that | X − Y | ≤ ε and Y is a discrete random variable. F Y ( y ) = Pr [ Y ≤ y ] = for 0 ≤ y ≤ 1 Random Variable: Y distance from center. 1 for y > 1  Also, if f X ( x ) = d dx F X ( x ) , then F X ( x + ε ) − F X ( x ) ≈ f X ( x ) ε . Probability within y of center: area of small circle Hence, Pr [ Y = n ε ] ≈ f X ( n ε ) ε . Pr [ Y ≤ y ] = Pr [ 0 . 5 < Y ≤ 0 . 6 ] = Pr [ Y ≤ 0 . 6 ] − Pr [ Y ≤ 0 . 5 ] area of dartboard Thus, we can think of X of being almost discrete with = F Y ( 0 . 6 ) − F Y ( 0 . 5 ) π y 2 Pr [ X = n ε ] ≈ f X ( n ε ) ε . = y 2 . = = . 36 − . 25 π Hence, = . 11  for y < 0 0  y 2 F Y ( y ) = Pr [ Y ≤ y ] = for 0 ≤ y ≤ 1  1 for y > 1 PDF. Target U [ a , b ] Example: “Dart” board. Recall that  0 for y < 0  y 2 F Y ( y ) = Pr [ Y ≤ y ] = for 0 ≤ y ≤ 1  1 for y > 1  0 for y < 0  f Y ( y ) = F ′ Y ( y ) = 2 y for 0 ≤ y ≤ 1 0 for y > 1  The cumulative distribution function (cdf) and probability distribution function (pdf) give full information. Use whichever is convenient.

Expo ( λ ) Random Variables A Picture The exponential distribution with parameter λ > 0 is defined by Continuous random variable X , specified by f X ( x ) = λ e − λ x 1 { x ≥ 0 } 1. F X ( x ) = Pr [ X ≤ x ] for all x . � 0 , Cumulative Distribution Function (cdf) . if x < 0 F X ( x ) = 1 − e − λ x , Pr [ a < X ≤ b ] = F X ( b ) − F X ( a ) if x ≥ 0 . 1.1 0 ≤ F X ( x ) ≤ 1 for all x ∈ ℜ . 1.2 F X ( x ) ≤ F X ( y ) if x ≤ y . � x − ∞ f X ( u ) du or f X ( x ) = d ( F X ( x )) 2. Or f X ( x ) , where F X ( x ) = . dx Probability Density Function (pdf). The pdf f X ( x ) is a nonnegative function that integrates to 1. � b Pr [ a < X ≤ b ] = a f X ( x ) dx = F X ( b ) − F X ( a ) The cdf F X ( x ) is the integral of f X . 2.1 f X ( x ) ≥ 0 for all x ∈ ℜ . � ∞ 2.2 − ∞ f X ( x ) dx = 1 . Recall that Pr [ X ∈ ( x , x + δ )] ≈ f X ( x ) δ . Think of X taking Pr [ x < X < x + δ ] ≈ f X ( x ) δ discrete values n δ for n = ..., − 2 , − 1 , 0 , 1 , 2 ,... with � x Pr [ X = n δ ] = f X ( n δ ) δ . Pr [ X ≤ x ] = F x ( x ) = − ∞ f X ( u ) du Note that Pr [ X > t ] = e − λ t for t > 0. Summary Continuous Probability 1 1. pdf: Pr [ X ∈ ( x , x + δ ]] = f X ( x ) δ . � x 2. CDF: Pr [ X ≤ x ] = F X ( x ) = − ∞ f X ( y ) dy . b − a 1 { a ≤ x ≤ b } ; F X ( x ) = x − a 1 3. U [ a , b ] : f X ( x ) = b − a for a ≤ x ≤ b . 4. Expo ( λ ) : f X ( x ) = λ exp {− λ x } 1 { x ≥ 0 } ; F X ( x ) = 1 − exp {− λ x } for x ≤ 0 . 5. Target: f X ( x ) = 2 x 1 { 0 ≤ x ≤ 1 } ; F X ( x ) = x 2 for 0 ≤ x ≤ 1.