EC381/MN308 Probability and Some Statistics. Lecture 7 - Outline. Chapter Cumulative Distribution Function (CDF) Continuous Random Variables

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1 EC38/MN38 Probability and Some Statitic Yanni Pachalidi Lecture 7 - Outline. Continuou Random Variable Dept. of Manufacturing Engineering Dept. of Electrical and Computer Engineering Center for Information and Sytem Engineering Chapter 3 Continuou Random Variable Continuou Random Variable = RV whoe range i a continuou ubet of the real number, i.e., take an uncountably infinite number of poible value 3. Cumulative Ditribution Function (CDF) The cumulative (probability) ditribution function of a random variable X (continuou or dicrete) i the total probability ma from - up to (and including) the point : F X () = P [X ], or F X () = P[X (-,]] Sample Space outcome Event a b Continuou Random Variable X R CDF (Dicrete RV) F X () P[ k ] F X ( ) CDF (continuou RV) P [a < X b] Key iue: Often cannot aociate a nonzero probability with any individual outcome Cannot enumerate eperiment outcome. Can only define probabilitie of event! Will focu on event repreenting outcome where random variable take value in interval 3 k a b P [a < X b] = F X (b) F X (a) 4 Propertie of the CDF - F X () i a non-negative real-valued function defined for all real number value of it argument (- < < ) - F X () [,], F X (- ) = and F X (+ ) = - F X () i a monotonic nondecreaing function of, i.e., if a < b, then F X (a) F X (b) Propertie of the CDF for continuou RV F X () i a continuou function of, i.e., F X (u) = F X (u + ) = F X (u ) P [X = u] =. Every ingle-value event {X = u} ha zero probability. Such an event i phyically unobervable ince it require an intrument with infinite preciion. P [X < ] = P [X ] = F X () Nonzero probabilitie are aigned to interval of the line. For a < b, P [a < X b] = F X (b) F X (a) F X () Continuou u u u + 5 Other propertie of the continuou CDF If i any number in the range < <, then there mut be at leat one number t uch that F X (t) = Intermediate value theorem: F X (- ) =, F X ( ) =, o a continuou function take all value between and F X () Can there be many t uch that F X (t) =? Thi could happen: there could be an interval of t uch that F X (t) = Mut be a ingle interval, becaue F X (t) i monotone non-decreaing. F X () t 6

2 Eample Chooe a random number between and, i.e., S = { : < < } Intuitively, the meaning of random in thi intance i that we do not favor any one number over other in the interval (,) One way of epreing the innate randomne of the choice i a follow: Given any ubinterval of (, ), the probability that the choen number lie in that ubinterval i equal to the length of that interval F X () Continuou but not differentiable Advanced topic: More general definition of continuou RV A continuou random variable X i one whoe CDF, F X (), i. continuou at all, < <. differentiable at all ecept poibly at a countable et of point < < n < More preciely, any finite-length interval contain at mot a finite number of point where F X () i not differentiable. The CDF of a dicrete random variable alo atifie thi condition. P.S. Thi i an eample of a continuou, but not everywhere differentiable, CDF 7 Note: The book doe not tre that there can be point where F X (u) i continuou but not differentiable 8 Mied (Continuou/Dicrete) RV Mied RV have piecewie differentiable CDF with poitive lope and jump dicontinuitie 3. Probability Denity Function (PDF) For a continuou RV X with a CDF F X () that i differentiable almot everywhere, the probability denity function (PDF) i the derivative of the CDF : CDF F X () F X () Dicrete RV Continuou RV Mied RV The PDF of a continuou RV i not a probability and may take value greater than one. It i a probability denity. However, the integral of a PDF over a region of i a probability. f X () PDF 9 Propertie of the PDF f X (), i.e., PDF i a non-negative function Proof: = area under the curve f X () from a to b. f X () i.e., the PDF ha unit area PDF f X (+ ) = ; f X ( ) =, i.e., a the argument tend to ±, the PDF curve mut decay away to (or ele the area under it would not be finite). The lope of the CDF F X () goe to a a b PDF veru PMF PMF of a Dicrete RV define a et of point mae on the ai: Total ma =. PMF P X () = ma at = probability that occur. PDF of a Continuou RV define a pread of the total probability ma of along the ai. There i no probability ma at any point. PDF of a continuou RV i not a probability. It provide the denity of the ma at each point The PDF i meaured in unit of probability ma/length The PDF i analogou to ma or charge denity, etc. If f X (a) i poitive at the point a and δ i the length of an interval, then P[a X a + δ ] f X (a) δ The approimation become better a δ become maller.

3 Eample A continuou random variable X ha PDF f X () =.75( ) for -, and otherwie. Compute P [.5 X.5]: Eample A continuou random variable X ha PDF f X () = for.5.5, and otherwie: Incorrect Incorrect Mot problem on continuou random variable are eaier to viualize with a diagram, which help in figuring out the limit and avoiding error. Correct anwer.75 Correct anwer f X () P [X.5] = area under PDF to left of.5 = haded area =.75 4 Eample 3 Finding the CDF from the PDF Eample 4 f X () =, for, =, otherwie f X () =.75( ) for, =, otherwie F X (.6) = P [X.6] = P [ < X.6] = area under PDF from to.6 = (/).6. = (.6) = Find F X ():.75 f X () F X (.3) = P [ X.3] = P [ < X.3] = area under PDF from to.3 = (/).3.6 = (.3) =.9 (increae a it hould).3.6 P [X ] = area under PDF curve to the left of = / by ymmetry! 5 6 Advanced topic: PDF for non-differentiable CDF Eample F X () f X () The derivative of the CDF of a continuou random variable X eit for almot all real number We are allowed to et the value of f X () to any nonnegative number only at thoe few iolated point where the CDF i not differentiable Furthermore, the arbitrarily choen value aigned to the pdf at thee iolated point make no difference whatoever in any probability calculation The probability that thi number occur i! /6 3 3 The derivative of the CDF i undefined at = +3 or - 3 Could chooe value a right derivative or left derivative, but doen t matter a long a value i nonnegative 7 8 3

differentiable at =. f X () ha value for < BUT f X () =?

e., X take on value in (, ) 9 Significance: If we repeat an eperiment N time, add up all oberved

center of gravity, Epectation of a Function of a Random Variable g(x) i a function of a continuou

variable: Eample Y = X, for X, Y = X, for X > Y = g() Moment and Central Moment Eample Quiz 3.

) n ] E [X μ X ] E [(X μ X ) ] Var[X] σ X = {Var[X]} / = mean (firt moment) of X = n-th moment of

central moment of X = econd central moment of X = Var[X] = Variance = E [X ] μ X (imilar to moment

4 Eample of convention F X () 3.3 Epectation Definition: The epected value (average) of a random variable X i CDF Not differentiable at =. f X () ha value for < BUT f X () =? Two convenient choice: Dicrete Continuou f X () = for, and elewhere i.e., X take on value in [, ] f X () = for < <, and elewhere i.e., X take on value in (, ) 9 Significance: If we repeat an eperiment N time, add up all oberved value of X, and divide by N, the reult will be pretty cloe to E [X] Center of probability ma, center of gravity, Epectation of a Function of a Random Variable g(x) i a function of a continuou random variable X g(x) i not necearily a continuou function By analogy with dicrete random variable: Eample Y = X, for X, Y = X, for X > Y = g() Moment and Central Moment Eample Quiz 3.3 Definition (ame a for dicrete random variable) μ X E [X] E [X n ] E [X a] E [(X a) n ] E [(X μ X ) n ] E [X μ X ] E [(X μ X ) ] Var[X] σ X = {Var[X]} / = mean (firt moment) of X = n-th moment of X = (firt) moment of X about a = n-th moment of X about a = n-th central moment of X = firt central moment of X = econd central moment of X = Var[X] = Variance = E [X ] μ X (imilar to moment of inertia around mean) = Standard deviation = pread around mean (imilar to radiu of gyration) RV Y, with PDF : = linpace(-,,); mak = find(( >= -)&( <= )); y = zero(ize()); y(mak) = 3*(mak).*(mak)/; plot(,y) 3 4 4

5 Epectation of Linear Function of a RV (ame a for dicrete cae) Mean Y = ax + b E [ax + b] = ae[x] + b The mean i multiplied by the ame factor a and hifted by the ame factor b. The epectation i a linear operation, i.e., the epectation of a weighted um i the weighted um of the epectation Variance Var[aX + b] = a Var[X] The variance i multiplied by the quare of a and i inenitive to the hift factor b. Proof Var[aX] = E [(ax) ] (E [ax]) = a E [X ] (aμ X ) = a (E [X ] (μ X ) ) = a Var[X] 5 Epectation cannot alway be defined: Since f X () for >, but f X () for <, E [X] i the difference between two poitive integral over (, ) and (, ). If both integral are infinite, E [X] i undefined. Thi doe not mean, however, that the PDF i not ueful (random fractal are often characterized by uch PDF). Eample The Cauchy RV ha a PDF f X () = [π(+ )] Thi PDF i ymmetric about the origin but ha long (power-law) tail. The integral of [(π(+ )] i of the form and E [X] i undefined. Some RV have finite mean but higher moment are undefined. 6 5

Source slideplayer.com/fundamentals of Analytical Chemistry, F.J. Holler, S.R.Crouch. Chapter 6: Random Errors in Chemical Analysis

Source slideplayer.com/fundamentals of Analytical Chemistry, F.J. Holler, S.R.Crouch. Chapter 6: Random Errors in Chemical Analysis Source lideplayer.com/fundamental of Analytical Chemitry, F.J. Holler, S.R.Crouch Chapter 6: Random Error in Chemical Analyi Random error are preent in every meaurement no matter how careful the experimenter.