# prv_lec2 - Probability Random Variables Chapter 2 Hyung Il...

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Probability & Random VariablesChapter 2Hyung Il Koo[email protected]Department of Electrical EngineeringAjou University, KoreaSpring 2016
2.0 NotationsCapital letters (e.g.W, X, Y, Z,· · ·): Random Variable (R.V.)Lowercase letters (e.g.w, x, y, z,· · ·): particular value of aR.V.S: sample space (whole set)
2.1 Definition of a Random VariableDefinitionR.V.: a realfunctionof the elements of a sample spaceS.i.e.,sSX(s)R.V.(real number)X:S: function that maps sample space to the realnumber space.Quite often we omit(s). i.e., instead ofX(s), we useX.Two conditions to be a R.V.1.x,{s:X(s)x}should be an eventP(Xx) =P({sS:X(s)x}).2.(Extreme points):P(X=−∞) =P(X=) = 0.
2.1 Discrete & Continuous R.V.’sDiscrete R.V.: pmf(probability mass function)Continuous R.V.: pdf(probability density function)
2.1 Definition of a Random VariableA sample spaceS={1,2,3,4}A random variableXis defined byX=X(s) =s3
2.2 Distribution FunctionP(Xx): probability of the event{s:X(s)x}.Depends onx.(Cumulative) probability distribution function (CDF):FX(x) =P(Xx)Properties of CDF1.FX(−∞) = 02.FX() = 13.0FX(x)1,x4.FX(x1)FX(x2)ifx1<x2;nondecreasing5.P(x1< Xx2) =FX(x2)FX(x1)6.FX(x+) =FX(x);rightcontinuouse.g. dice rolling
2.3 Density FunctionPDF(probability density function):fX(x)- derivative ofCDF.fX(x) =dFX(x)dxSometimes just calldensity functionFor discrete variable, the termPMFis more used.Continuous & differentiablefX(x
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