6 chapter8 Random Variate

3 figure inversetransform technique for exp 1 4

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Unformatted text preview: Figure: Inversetransform technique for exp( = 1) 4 Exponential Distribution [Inverse-transform] Example: Generate 200 variates Xi with distribution exp( = 1) Generate 200 Rs with U(0,1) and utilize eq’n 8.3, the histogram of Xs become: Check: Does the random variable X1 have the desired distribution? P( X 1 x0 ) = P( R1 F ( x0 )) = F ( x 0 ) 5 Other Distributions [Inverse-transform] Examples of other distributions for which inverse cdf works are: Uniform distribution Weibull distribution Triangular distribution 6 Empirical Continuous Dist’n [Inverse-transform] When theoretical distribution is not applicable To collect empirical data: Resample the observed data Interpolate between observed data points to fill in the gaps For a small sample set (size n): Arrange the data from smallest to largest x (1) x (2) … x (n) Assign the probability 1/n to each interval ˆ X = F 1 ( R) = x(i where ai = x(i ) x(i 1) 1 / n (i 1) / n x (i-1) x x (i) (i 1) 1) + ai R n = x(i ) x(i 1) 1/ n 7 Empirical Continuous Dist’n [Inverse-tr...
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