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RandomVariables-Densities-Distributions

# RandomVariables-Densities-Distributions - Random Variables...

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1 Copyright F. Novomestky 2007 Random Variables, Densities, Distributions ¾ This supplement provides a catalog of some of the most widely used random variables in financial risk modeling. ¾ Extensive use of the Dirac delta function, unit step function and ramp function. ¾ Additional comments on their use. ¾ Some are based on current research. 2 Copyright F. Novomestky 2007 Random Variables, Distributions, Densities Uniform Random Variable A special case of the previous example is uniform distribution defined on the interval [0,1] () 00 10 1 01 U u fu u u < = < ⎪⎩ 3 Copyright F. Novomestky 2007 Random Variables, Distributions, Densities Uniform Random Variable (continued) We can represent this distribution using the unit step function 1 U f uH u = −− The distribution function is obtained by integrating the density function 1 u UU Fu ftd t Ru −∞ = =

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4 Copyright F. Novomestky 2007 Random Variables, Distributions, Densities () 00 01 11 U u Fu u u u < = < ⎪⎩ 5 Copyright F. Novomestky 2007 Random Variables, Distributions, Densities General Uniform Random Variable Suppose that X is a uniformly distributed random variable in the interval [a,b] ()() 1 X f xH x a H x b ba ⎡⎤ =− ⎣⎦ The corresponding distribution function is 1 U F uR u a R u b 6 Copyright F. Novomestky 2007 Random Variables, Densities, Distributions x 2 f(x) x 1 h s 2 1 s 1 1 Triangle Random Variable
7 Copyright F. Novomestky 2007 Random Variables, Densities, Distributions Triangle Random Variable The random variable has a worst case value of zero, a best case value of x 2 and a most likely value of x 1 . () 11 2 1 1 2 2 00 0 0 X x sx x x fx s xs xxxxx xx ≤≤ = + −≤ 8 Copyright F. Novomestky 2007 Random Variables, Densities, Distributions Triangle Random Variable Expressions for h, s 1 , and s 2 are easily derived from analytic geometry.

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RandomVariables-Densities-Distributions - Random Variables...

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