Continuous Random Variables
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1 CPSC 53 Systems Modeling nd Simultion Continuous Rndom Vriles Dr. Anirn Mhnti Deprtment of Computer Science University of Clgry Definitions A rndom vrile is sid to e continuous if there eists non-negtive function f(), (, ), with the property tht for ny set A of rel numers: P ({ A}) f ( ) d A f() is clled the proility density function (PDF) of Continuous Rndom Vrile Properties of PDF f ( ), f ( ) d }) f ( ) d i. e., B [, ] nd we wnt to find i.e., re under the curve equls one. P{ B}. Continuous Rndom Vrile 3
2 Properties of PDF (continued) Continuous distriutions ssign vlue to individul vlues: }) f ( ) d Consequence of the ove property: }) < < }) < }) < }) Continuous Rndom Vrile 4 Cumultive Distriution Function The CDF F ( ) of continuous rndom vrile with PDF f ( ) cn e otined s follows: F ( ) (, ]}) f ( t) dt Continuous Rndom Vrile 5 CDF - PDF Reltionship The PDF cn e otined from the CDF nd vice vers: F df ( ) ( ) d ' f ( ) Distriution of continuous rndom vrile cn e represented using either the PDF or the CDF. Continuous Rndom Vrile 6
3 The Uniform Distriution A rndom vrile is sid to e uniformly distriuted on the intervl [, ], <, if the proility of selecting point long the intervl [, ] is eqully likely t ll portions of the intervl. We cn sy tht the proility tht will elong to prticulr su-intervl of [, ] is proportionl to the length of tht su-intervl. Continuous Rndom Vrile 7 PDF nd CDF of Uniform R.V. The PDF of uniform rndom vrile in the intervl [, ] is:, < < f ( ), otherwise How did we get F()? F ( < < ) dt The CDF of is:, F ( ), < <, dt Continuous Rndom Vrile 8 Uniform R.V. PDF nd CDF PDF of Uniform R.V. (, 3) CDF of Uniform R.V. (, 3) f().5 F() Continuous Rndom Vrile 9 3
4 Eponentil Distriution A continuous rndom vrile is eponentilly distriuted with prmeter β if it hs the following PDF: f β e ( ), The CDF for the eponentil distriution is: F β, ( ) e, β, otherwise < Continuous Rndom Vrile Eponentil Models This distriution hs een used to model: Inter-rrivl times etween IP pckets Inter-rrivl times etween clls t cll centre Inter-rrivl times etween we sessions from we client Service time distriutions Lifetime of products Widely used in queuing theory Continuous Rndom Vrile Eponentil PDF nd CDF PDF of Eponentil Distriution CDF of Eponentil Distriution 4 3 f() β.5 β. β. β F() β Continuous Rndom Vrile 4
5 Memory-less Property of Eponentil Distriution Suppose inter-rrivl times of IP pckets re modelled using Eponentil distriution. The memory-less property sttes tht the distriution of the epected time to pcket rrivl is independent of the durtion there hve een no pcket rrivls Suppose is n eponentilly distriuted r.v. nd t (i.e., no rrivls for time t or less). Then, t+h t }) h }) Continuous Rndom Vrile 3 Memory-less Property of Eponentil Distriution (cont.) Regrdless of the durtion of no pcket rrivls, the proility tht n rrivl will occur in the net h time units remins the sme. t + h}) t + h t}) t}) β ( t+ h) [ e ] β h e β t [ e ] h}) Eponentil distriutions re the only continuous distriutions with the memory-less property. Continuous Rndom Vrile 4 Norml Distriution is norml rndom vrile with men μ nd vrince if hs the following PDF: f ( ) The CDF of norml distriution is: There is no closed form for F (). π e ( μ), F ( ) }) < < e π ( t μ) dt Continuous Rndom Vrile 5 5
6 PDF of Norml Distriution PDF of Norml Distriution (μ ).8.6 f() This PDF hs ell shpe with pek t. Continuous Rndom Vrile 6 Why is Norml Distriution Importnt? Approprite or useful for modelling mny relworld phenomen IQs of rndomly selected people Height of people selected from homogeneous popultion Mthemticl convenience: mny sttisticl nlysis methods ssume norml distriution Centrl Limit Theorem: The smple men of lrge, rndom smple from ny distriution with finite vrince will e pproimtely normlly distriuted. Continuous Rndom Vrile 7 More on Norml Distriution Terminology: lso clled the Gussin distriution, fter Germn mthemticin nd scientist Crl Friedrich Guss ( ) In norml distriution 68% of the vlues re within stndrd devition of the men Approimtely 95% of the vlues re within stndrd devitions of the men Approimtely 99% of the vlues re within 3 stndrd devitions of the men Continuous Rndom Vrile 8 6
7 Even more on Norml Distriution If is normlly distriuted with prmeters μ nd, then ( μ) Z is normlly distriuted with prmeters nd. Z is clled the stndrd norml distriution. Continuous Rndom Vrile 9 Computing CDF of Norml Distriution Norml distriution hs no closed form for F(). How to compute < < })? Trnsform to stndrd norml distriution Z nd use tles ( μ) If ~ N( μ, ) then Z is N(,) Continuous Rndom Vrile Omnipresence of Norml Distriution! Norml distriution is everywhere, even on nk note. Continuous Rndom Vrile 7
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