NOTES ON DISTRIBUTIONS MICHAEL N KATEHAKIS Radom Variables Radom variables represet outcomes from radom pheomea They are specified by two objects The rage R of possible values ad the frequecy fx with which values from withi the rage ca occur Whe the rage is a discrete set we have a discrete radom variable ad whe the rage is cotiuous we have a cotiuous radom variable See several examples below For otatioal coveiece we ofte exted the rage to a larger set where outcomes outside the origial rage are assiged zero frequecy The terms: a radom variable X, the distributio fx, or the Rage R, is discrete respectively cotiuous are equivalet The cumulative frequecy or distributio fuctio, of a radom variable X, is defied as F x = { x fy if X is discrete, x fydy if X is cotiuous Expectatio ad momets The expected value or first momet of a radom variable X is a costat EX defied by { xfx if X is discrete, EX = xdf x = xfxdx if X is cotiuous R The th momet of a radom variable X is a costat EX defied by { EX = x df x = x fx if X is discrete, x fxdx if X is cotiuous R The variace of a radom variable X is a costat varx or σ X defied by V arx = E[X EX ] 3 Joit Distributios - Idepedet radom variables Ofte two radom variables X ad Y, have a joit distributio defied by a two dimesioal frequecy fuctio fx, y I the discrete case I the cotiuous case fx, y = P X = x, Y = y P X x, Y y = x x The margial frequecy, i the discrete case, is defied fx = P X = x = y= fx, y dx dy fx, y I the cotiuous case it is easier to defie the margial distributio:
MICHAEL N KATEHAKIS F x = P X x = y= fx, y The covariace of two joitly distributed radom variables X ad Y is a costat CovX, Y defied by CovX, Y = E[X EXY EY ] Two joitly distributed radom variables X ad Y are idepedet if: F x, y = F xf y for all x, y Properties expectatio ad variace EaX + b = aex + b, EX + Y = EX + EY, EXY = EXEY, for idepedet X ad Y, V arx = EX EX, V arax + b = a V arx, V arx + Y = V arx + V ary, for idepedet X ad Y, V arx + Y = V arx + V ary + CovX, Y, CovX, Y = EXY EXEY, CovX, X = varx Summary of EX ad V arx for various distributios for X X EX V arx Beroullip p p p Biomial, p p p p Geometricp p P oissoλ λ λ Uiform{,, } + a+b p p Uiforma, b Nµ, σ µ σ expλ λ λ b a 4 Discrete radom variables The Beroulli radom variable represets experimets trails with oly two possible outcomes - success or failure 0 We write X Beroulli f Rage fx 0 f0 f where P X = = f ad P X = 0 = f0 = f
NOTES ON DISTRIBUTIONS 3 The Biomial radom variable represets the umber of successes i idepedet Beroulli trials each with success probability p = f We write: X Biomial, p Rage fx 0 f0 = 0 f = f = f = f = p p p p p p p p Note that p is the probability of successes, p is the probability of failures, is the umber of outcomes with successes i Beroulli trials Applicatios: # of defectives i a sample # of successful requests # of days without a accidet, etc Relatio with Beroulli: Biomial, p = Beroullip, ie, a Beroulli rv is the # of success i a sigle Beroulli trial, if X, X,, X, are idepedet Beroullip, the Y = i= X i Biomial, p 3 The Geometric radom variable represets the umber of trials util you get the first success i idepedet Beroulli trials each with success probability p We write: X Gp Rage fx f = p f = f = p p p p Memoryless Property A importat property of the geometric distributio is: P X > s + t X > t = P X > s for all s, t = 0,, This property meas that if you repeat a experimet util the first success, the, give that the first success has ot yet occurred, the coditioal probability distributio of the umber of additioal trials does ot deped o how may failures have bee observed The die oe throws or the coi oe tosses
4 MICHAEL N KATEHAKIS does ot have a memory of these failures The geometric distributio is the oly memoryless discrete distributio Applicatios # of tests to locate a bug # waitig for a evet arrival of a customer i discrete time, etc Example Suppose a computer program has a error i it To locate the error, ivestigators coduct a series of idepedet tests Each test is based o radom iputs ad it detects the error with probability 030 a The the probability of detectig the error o the third test is 7 3 b The probability that it taes at least three tests to detect the error is P X 3 = P X = + P X = + P X = 3 = 3 + 7 3 + 7 3 4 The Poisso radom variable typically represets the umber of evets that occur i a fixed time iterval, whe two evets are uliely to occur simultaeously durig a short time period We write: X P oissoλ The parameter λ represets the rate, ie, the average umber of evets per uit of time or area Rage fx 0 f = e λ f = e λ λ f = e λ λ! Applicatios # of customer arrivals i a store durig a day, # of job arrivals durig a fixed period of time, # of accidets, # of telephoe calls i discrete time, etc Relatio with Biomial: Whe i a Biomial p is small p 005 ad is large geq0 ie, the successes are rare, ad the umber of trials i large, the P oissop = Biomial, p We use this Poisso approximatio for Biomial wheever possible as worig with the Poisso is easier 5 The Discrete uiform radom variable typically represets outcomes from a fiite set that each ca occur with equal probability Rage fx f = / f = / f = / Applicatios the demad realized i a fixed time period, whe throwig a fair dice the possible values are,, 3, 4, 5, 6, ad each time the dice is throw, the probability of a give score is /6, etc
NOTES ON DISTRIBUTIONS 5 5 Cotiuous radom variables 6 The Expoetial radom variable typically represets the radom time betwee evets that happe at a costat average rate We use the otatio X expλ Rage fx F x [0, ] λe λx λe λx The expoetial distributio may be viewed as a cotiuous couterpart of the geometric distributio, which describes the umber of Beroulli trials ecessary for the first success The expoetial distributio models the time util the first arrival of a Poisso processes Specifically, suppose that Nt =# of rare evets i [0, t] P oissoλt Let X = time of first evet or X = time of ext evet, or X = time betwee two evets The, X expλ Memoryless Property A importat property of the expoetial distributio is: P X > s + t X > t = P X > s for all s, t 0 This meas, for example, that the coditioal probability that we eed to wait, 5 more secods before the first arrival, give that the first arrival has ot yet happeed after 30 secods, is the same as the iitial probability that we eed to wait more tha 5 secods for the first arrival ie, P X > 35 X > 30 = P X > 5 The expoetial distributio is the oly memoryless cotiuous distributio Applicatios the iter-arrival times i queues, the time it taes to process a order at a fast-food restaurat, etc 7 The Uiform radom variable typically represets outcomes such that values i all itervals of the same legth are equally probable We use the otatio X Ua, b, where b > a The mai applicatio of uiform rvs is i the simulatio of radom variables Rage fx F x a, b /b a x a/b a Stadard Uiform distributio: X U0, 8 The Normal radom variable typically represets outcomes such that values i itervals close to the mea are more probable tha values i itervals of the same legth that are far from the mea It is ofte called the bell curve because the graph of its probability desity resembles a bell The stadard ormal distributio is the ormal distributio with a mea of zero ad a stadard deviatio of oe; its desity ad its distributio is deoted with φx ad P hix, respectively We use the otatio X Nµ, σ Rage fx F x, + σ x µ e σ π x fydy Some properties of the Normal distributio a P µ σ < X < µ + σ = 68, b P µ σ < X < µ + σ = 95, c P µ 3σ < X < µ + 3σ = 997, If X Nµ, σ, ad Y = a + bx the Y Na + bµ, b σ 3 If X Nµ, σ, ad Y = X µ/σ the Y N0,
6 MICHAEL N KATEHAKIS Refereces Averill M Law ad W David Kelto 999 Simulatio Modelig ad Aalysis 3rd ed McGraw-Hill Higher Educatio, New Yor, NY Mauel Lagua ad Joha Marlud 004 Busiess Process Modelig, Simulatio, ad Desig, Pretice Hall, Eglewood Cliffs, NJ Departmet of Maagemet Sciece ad Iformatio Systems, Rutgers Busiess School, Newar ad New Bruswic, 80 Uiversity Aveue, Newar, NJ 070-895 E-mail address: m@rcirutgersedu