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Get quantile function of dynamic mixture model

I have a dynamic mixture distribution fitted to my risk data (i.e., I have all parameters) of Weibull and Generalized Pareto, with a Cauchy CDF mixing function, that can be written as: \begin{align} ...
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How to sample from a distribution so that mean of samples equals expected value?

Given a random variable $X$, how do I have obtain $N$ random variates of $X$ so that the mean value of my samples equals the expected value of $X$? E.g. let $X$ have uniform distribution on the ...