Distributions. (corresponding to the cumulative distribution function for the discrete case).


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1 Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive distribution function for the discrete cse). Smpling from the distribution corresponds to solving the eqution = for given rndom probbility vlues.   f(z)dz f(z)dz I. Uniform Distribution probbility density function (re under the curve = ) p() (b) b The pdf for vlues uniformly distributed cross [,b] is given by f() = Smpling from the Uniform distribution: (b) (pseudo)rndom numbers drwn from [,] distribute uniformly cross the unit intervl, so it is evident tht the corresponding vlues = + (b) slope = (b) will distribute uniformly cross [,b]. Note tht directly solving =  for s per = f(z)dz b  dz = z b  b   b  lso yields = + (b) (ectly s it should!). = b
2 The men of the uniform distribution is given by b b b + μ = E(X) = z dz = = (midpoint of [, b] ) b  b  z f(z) dz The stndrd devition of the uniform distribution is given by σ = E((X μ) b b + (b  ) ) = z  dz = (with some work!) b  (zμ) f(z) dz II. Norml Distribution For finite popultion the men (m) nd stndrd devition (s) provide mesure of verge vlue nd degree of vrition from the verge vlue. If rndom smples of size n re drwn from the popultion, then it cn be shown (the Centrl Limit Theorem) tht the distribution of the smple mens pproimtes tht of distribution with men: μ = m stndrd devition: pdf: σ = f() = e σ π s n (μ) σ which is clled the Norml Distribution. The pdf is chrcterized by its "bellshped" curve, typicl of phenomen tht distribute symmetriclly round the men vlue in decresing numbers s one moves wy from the men. The "empiricl rule" is tht pproimtely 68% re in the intervl [μσ,μ+σ] pproimtely 95% re in the intervl [μσ,μ+σ] lmost ll re in the intervl [μ3σ,μ+3σ] This sys tht if n is lrge enough, then smple men for the popultion is ccurte with high degree of confidence, since σ decreses with n. Wht constitutes "lrge enough" is lrgely function of the underlying popultion distribution. The theorem ssumes tht the smples of size n which re used to produce smple mens re drwn in rndom fshion. Mesurements bsed on n underlying rndom phenomen tend to distribute normlly. Hogg nd Crig (Introduction to Mthemticl Sttistics) note tht the kinds of phenomen tht hve been found to distribute normlly include such disprte phenomen s ) the dimeter of the hole mde by drill press, ) the score on test, 3) the yield of grin on plot of ground, 4) the length of newborn child.
3 The ssumption tht grdes on test distribute normlly is the bsis for soclled "curving" of grdes (note tht this ssumes some underlying rndom phenomen controlling the mesure given by test; e.g., genetic selection). The prctice could be to ssign grdes of A,B,C,D,F bsed on how mny "stndrd devitions" seprtes percentile score from the men. Hence, if the men score is 77.5, nd the stndrd devition is 8, then the curve of the clss scores would be given by A: 94 nd up (.5%) B: (3.5%) C: 785 (68%) D: 669 (3.5%) F: otherwise (.5%) Most people "pss", but A's re hrd to get! This could be pretty distressing if the men is 95 nd the stndrd devition is (i.e., 9 is n F). A plot of the pdf for the norml distribution with μ = 3 nd σ = hs the ppernce: f() μ σ Note tht the distribution is completely determined by knowing the vlue of μ nd σ.
4 The stndrd norml distribution is given by μ = nd σ =, in which cse the pdf becomes f() = e π nsmple = nsmple z e π dz nsmple.5 It is sufficient to smple from the stndrd norml distribution, since the liner reltionship = μ + σ nsmple substitute z=µ+σ t holds. dz = σdt σ (zμ ) μ+ σ nsmple (μ+ σtμ ) μ+ σ nsmple t  σ e π dz =  e σ π σdt =  e π There is no "closedform formul" for nsmple, so pproimtion techniques hve to be used to get its vlue. dt
5 III. Eponentil Distribution The eponentil distribution rises in connection with Poisson processes. A Poisson process is one ehibiting rndom rrivl pttern in the following sense:. For smll time intervl Δt, the probbility of n rrivl during Δt is Δt, where = the men rrivl rte;. The probbility of more thn one rrivl during Δt is negligible; 3. Interrrivl times re independent of ech other. [this is kind of "stochstic" process, one for which events occur in rndom fshion]. Under these ssumptions, it cn be shown tht the pdf for the distribution of interrrivl times is given by  f() = e which is the eponentil distribution. More to the point, if it cn be shown tht the number of rrivls during n intervl is Poisson distributed (i.e., the rrivl times re Poisson distributed), then the interrrivl times re eponentilly distributed. Note tht the men rrivl rte is given by nd the men interrrivl time is given by /. The Poisson distribution is discrete distribution closely relted to the binomil distribution nd so will be considered lter. It cn be shown for the eponentil distribution tht the men is equl to the stndrd devition; i.e., μ = σ = / Moreover, the eponentil distribution is the only continuous distribution tht is "memoryless", in the sense tht P(X > +b X > ) = P(X > b). f()  f() = e
6 When =, the distribution is clled the stndrd eponentil distribution. In this cse, inverting the distribution is strightforwrd; e.g., nsmple z e dz =  e z nsmple nsmple = log e () nsmple = log e () =  e nsmple which is closed form formul for obtining normlized smple vlue (nsmple) using rndom probbility. Generl smple vlues () cn then be obtined from the stndrd eponentil distribution by = nsmple =  log e = ( ) log ( )  e / (e)/e The evident utility of the eponentil distribution in discrete systems simultion is its effectiveness for modeling the rndom rrivl pttern represented in Poisson process. Smpling for interrrivl times is nturl pproch for introducing new items into the model one t time. However, cre must be tken tht when used for this purpose, the eponentil distribution is pplied to reltively short time periods during which the rrivl rte is not dependent on time of dy (for emple, the model could progress in hour service intervls representing slow, moderte, nd pek demnd, ech governed by n eponentil distribution with n pproprite men interrrivl time). A more sophisticted pproch is to djust the rrivl rtes dynmiclly with time, concept studied under the topic of joint probbility distributions, which will be discussed lter.
7 To verify tht μ = σ = /, integrte by prts to obtin ech of μ nd σ s follows:  z z z μ = E(X) = z e dz =  ze + e dz = + e = u dv uv  vdu σ = E z z z ( X /) ) = ( z /) e dz =  ( z /) e   e ( z /)dz z = + z e dz  u dv e z dz = = / = (by definition) For the pdf of the eponentil distribution so f() = nd f () =   f() = e note tht f () =  e  Hence, if < the curve strts lower nd fltter thn for the stndrd eponentil. The symptotic limit is the is.
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