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Notes02 - The Concept of a Random Variable Random variables...

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1 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 The Concept of a Random Variable Random variables; distributions; densities Examples of distributions and density functions Conditional distributions and densities 2 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Random Variables; Distributions; Densities We are given the probability space E = ( S , F ,P) whose outcomes ξ S . For each ξ S , we have a rule that assigns a number X( ξ ), {X x} = { ξ | X( ξ ) x} {x 1 X x 2 } = { ξ | x 1 X( ξ ) x 2 } {X = x} = { ξ | X( ξ ) = x} {X I } = { ξ | X( ξ ) I } 3 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Random Variables; Distributions; Densities Example of a Random Variable S = {f 1 , f 2 , f 3 , f 4 , f 5 , f 6 } P(f i ) = 1/6 for all f i S X(f i ) = 10 i X(f 1 ) = 10, X(f 2 ) = 20, X(f 3 ) = 30 X(f 4 ) = 40, X(f 5 ) = 50, X(f 6 ) = 60
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4 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Random Variables; Distributions; Densities Example of a Random Variable {X 35} = {f i | X(f i ) 35} = {f 1 , f 2 , f 3 } {X 5} = 0 {20 X 35} = {f i | 20 X(f i ) 35} = {f 2 , f 3 } {X = 40} = {f 4 } 5 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Random Variables; Distributions; Densities Formal Definition A real random variable X is a real-valued function whose domain is S such that 1. The set {X x} is an event for any real number x, that is, X( ξ ) is measurable 2. P{X = + } = P{X = - } = 0 The distribution function of the r.v. X is F X (x) = P{X x} defined for x (- ,+ ) 6 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Random Variables; Distributions; Densities Number of RV’s Univariate Multivariate Type of RV’s Discrete Continuous Description Cumulative Distribution (CDF or DF) Non-cumulative Probability density function
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7 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Random Variables; Distributions; Densities Example of a Random Variable ξ P( ξ ) X( ξ ) x {X x} F X (x) f 1 1/6 10 0 0 0 f 2 1/6 20 10 {f 1 } 1/6 f 3 1/6 30 20 {f 1 ,f 2 } 2/6 f 4 1/6 40 30 {f 1 ,f 2 ,f 3 } 3/6 f 5 1/6 50 40 {f 1 ,f 2 ,f 3 ,f 4 } 4/6 f 6 1/6 60 50 {f 1 ,f 2 ,f 3 ,f 4 ,f 5 } 5/6 60 {f 1 ,f 2 .f 3 .f 4 ,f 5 ,f 6 } 6/6 8 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Random Variables, Distributions, Densities Distribution Function 0.00 0.20 0.40 0.60 0.80 1.00 1.20 0 20 40 60 80 100 120 x F(x) 9 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Random Variables, Distributions, Densities Properties of Distribution Functions a. F(- ) = 0, F( ) = 1 b. F(x 1 ) F(x 2 ) for x 1 x 2 c. Continuous from the right F(x+) = F(x)
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10 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Random Variables, Distributions, Densities Density Function The derivative of F(x) with respect to (w.r.t.) x ( ) ( ) dF x f x dx = 11 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Random Variables, Distributions, Densities Random Variables of the Continuous Type From Property b above f(x) 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) 2 1 2 1 2 1 1 2 1 x x x x x f x dx F F f x dx F x F F x F x F x f x dx P x X x f x dx −∞ −∞ = ∞ − −∞ = = −∞ = = = 12 FE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007
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