I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution.

Ideally, this adjustment would be implemented through a parametric function which, **multiplied by the pdf or cdf** of the original distribution, produces the desired mean or dispersion shift.

Is there any adjustment (even if approximate) that may achieve this?

(A possible solution could involve *Probability Weighting Functions*, but I am not sure if this approach can deliver the required mean or variance shift)

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#### Best Answer

To change the mean and variance, consider the following equations (scaling and shifting.)

**Scaling**: multiplying $x$ by constant $c$

$$ mean(cx)=c times mean(x)=cmu\ std(cx)=sqrt{c^2 var(x)}=sqrt{c^2sigma^2}=csigma $$

**Shifting**: adding constant $c$ to $x$

$$ mean(x+c)=mean(x) + c =mu + c\ std(x+c)=sqrt{var(x) + var(c)} = sqrt{sigma^2+0}=sigma $$

Update: to keep the mean unchanged while scalling the variance/std, follow the following steps

Let the $x$ is scaled by $c$, which gives $cmu$ and $csigma$. Then to scale back the mean to $mu$, shift it by $k = (1-c)mu$. The final distribution will have $mu_{new}= mu$, and $sigma_{new}= csigma$