Transcription of Random Variables and Measurable Functions.
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Random Variables and Measurable Functions.
Chapter 3 Random Variables andMeasurable MeasurabilityDefinition 42( Measurable function) Letfbe a function from a measurablespace( ,F)into the real numbers. We say that the function ismeasurableiffor each Borel setB B,theset{ ;f( ) B} 43( Random variable ) Arandom variableXis a Measurable func-tion from a probability space( ,F,P)into the real numbers<.Definition 44(Indicator Random Variables ) For an arbitrary setA FdefineIA( )=1if Aand0otherwise. Thisiscalledanindicator 45(Simple Random Variables ) Consider eventsAi F,i=1,2,3, .., nsuch that ni=1Ai= .DefineX( )=Pni=1ciIAi( )whereci <. ThenXis Measurable and is consequently a Random variable . We normally assumethat the setsAiare disjoint. Because this is a Random variable which cantake onlyfinitely many different values, then it is calledsimpleand any randomvariable taking onlyfinitely many possible values can be written in this 46(binomial tree) A stock, presently worth $20, can increase eachday by $1 or decrease by $1.
3.2. CUMULATIVE DISTRIBUTION FUNCTIONS 17 2. If X is a real-valued random variable then [X = −∞]=ϕthe empty set. Therefore for any sequence x
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