RandomNumbers

To generate numbers that follow wx therefore we draw

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Unformatted text preview: et function from -∞ to x (or the CDF until x, F (x)) is evaluated, we solve for x yielding a new equation x = F −1 (r). To generate numbers that follow w(x) therefore, we draw from the uniform distribution and we apply the transformation x = F −1 (r). For example, lets assume we would like to generate random numbers that follow the 1 exponential distribution w(x) = λ e−x/λ for x ≥ 0 and w(x) = 0 otherwise. Following the recipe from above x1 r= e−x /λ dx = 1 − e−x/λ 0λ Solving for x x = −λ ln(1 − r) v 4.1 Example The following code will produce numbers that follow the exp (−x) distribution. Figure 3 shows the resulting histogram of the generated numbers compared to the actual exp (−x). Listing 2: Inverse Transformation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 4.2 % AM207 , Spring 2013 % Pavlos Protopapas , Harvard University % INVERSE TRANSFORM FOR w ( x ) = e ( - x ) from 0 to 6 clear ; xmin = 0; xmax = 6; % probability distribution wanted w = @ ( x ) exp ( - x ) ; % The CDF of w until x F_w = @ ( x ) 1 - exp ( - x ) ; % solve for x by inverting the CDF invF_w = @ ( r ) - log (1 - r ) ; % Getting the limits rmin = F_w ( xmin ) ; rmax = F_w ( xmax ) ; % N = 10000; % Creating a rescaled uniform random number vector % and transforming R = ( rmax - rmin ) * rand (1 , N ) + rmin ; X = invF_w ( R ) ; clf hold on h = hist (X ,100) ; % plot histogram hist (X , 100) x = xmin :0.1: xmax ; % Plot distribution normalized to histogram plot (x , h (1) * w ( x ) , ’r - ’) Scaling Assume you want to produce numbers following 1/(x + 1) distribution. This can be inverted as x 1 r= dx = ln(x + 1) ∞ x +1 HOWEVER if r ∼ U [0, 1] then x will take values in the range [0, e − 1] vi Figure 3: Numbers generated using inverse transform for e−x P(z)= exp(−z) 600 500 400 300 200 100 0 0 1 2 3 z 4 5 6 To generate values x, from xmin to xmax we can generate r ∼ U [rmin , rmax ] where rmin = F (xmin ) and rmax = F (xmax ). In the example above F (x) = ln(x + 1) and therefore umin = ln(xmin + 1) and umax = ln(xmax + 1)...
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This document was uploaded on 10/07/2013.

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