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allelopath

reverse a probability distribution function

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(Sorry for the previous untitled post) I'm trying to figure out how to 'reverse' a probability distribution function. In particular a 2-parameter Weibull pdf. (see http://www.weibull.com/LifeDataWeb/weibull_probability_density_function.htm) I have written a java method which implements a distribution function (see java code below) So if I call the method with this input: K = 2.851300069247714 C = 7.596157491862076, (these values derived from the data set) and a U =7, it returns 0.14612429022044235 This means that there is a ~14% chance of getting a 7 What i really need is the reverse where it returns a value that would be likely based on the distrubution, similar to if i called java.lang.Math.rand(); How do i do this, mathematically? /** * calculateWeibull * * calculates the weibull distribution * * @param U * @param K shape factor * @param C scale factor * * @return Returns the value of the distribution at U */ public double calculateWeibull (double U, double K, // shape double C) // scale { double Pudu = 0.0; double firstHalf = 0.0; double secondHalf = 0.0; firstHalf = (K/C) * Math.pow(U/C, K-1); secondHalf = Math.exp( (-Math.pow(U/C, K))); Pudu = firstHalf * secondHalf; return Pudu; }

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A simple way is to calculate a random value r1 uniformly distributed in [0,1] and another value r2 uniformly distributed over valid values of U e.g maybe 1,2,..10 in your example. If r1<pdf(r2) then return r2 else repeat. It won't be very fast, but it'll work.

However, your U seems to be continuous and then P(U=U1) for any U1 is 0. You have to integrate over an interval to get a probability, i.e P(U in [U1,U2])=integral(U1,U2)[ pdf(x) dx ].

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Well, it can be nontrivial (i don't know this transformtion). But you should at least understand what you are doing.

You have a random variable X in R(0,1) (that is uniformly distributed in [0,1], it can be measured with the rand() function). And you want to convert this random variable into a weibull distribution.

For example, to transform a uniformly distributed varible into a normal distribution a so called Box-Muller transformation can be done:
http://mathworld.wolfram.com/Box-MullerTransformation.html

But as I said, the transformation from a uniformly distributed variable to a weibull distr. can be very hard and perhaps it can only be done numerically.

Regards

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