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- Given the support [a, b] of a probability density function. How can I change the formula for the probability density function with a support [u, v]?

Given the support [a, b] of a probability density function. How can I change the formula for the probability density function with a support [u, v]? Example: Given the beta distribution with support [a=0,b=1]:

$$\frac{x^{p-1} (1-x)^{q-1}}{Beta(p,q)}$$

Then the beta distribution with support [u,v] is given by

$$\frac{(x-u)^{p-1} (v-x)^{q-1}}{Beta(p,q) (b-a)^{p+q-1}}$$

My question is how should you make the transformation for an arbitrary probability density function?

$$\frac{x^{p-1} (1-x)^{q-1}}{Beta(p,q)}$$

Then the beta distribution with support [u,v] is given by

$$\frac{(x-u)^{p-1} (v-x)^{q-1}}{Beta(p,q) (b-a)^{p+q-1}}$$

My question is how should you make the transformation for an arbitrary probability density function?