Solved – How to modify the mean and variance/dispersion of a given distribution

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)

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 $sigma_{new}= csigma$

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