# How to change the support of a probability density function?

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In summary, the nature of transforming the support of a probability density function involves shifting and stretching the interval. To make the transformation for an arbitrary probability density function, you can define the variable y = (x-u)/(v-u) and use the equation f(x) = f(y)/(v-u) to replace x with y in the original equation. This results in a new probability density function with the desired support. The transformation of the support is done by using a transformation of coordinates, where the coefficients A and B are determined by the conditions T(a) = u and T(b) = v.

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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?

What is the nature of the transformation - simply shifting and stretching the interval? If so, the answer is yes, if done right.

Yes, the nature of the transformation is simply shifting and stretching the interval, but my question was how should you make the transformation for an arbitrary probability density function?

Define the variable y = (x-u)/(v-u), so 0≤y≤1
For slices of equal area, f(y)dy = f(x)dx
Hence f(x) = f(y).dy/dx = f(y)/(v-u)
Replace x by y in your first equation, and you get f(x) as your second equation. (I assume you meant (v-u) in the denominator, not (a-b).)

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}}##

did you mean ##\frac{(x-u)^{p-1} (v-x)^{q-1}}{Beta(p,q)(v-u)^{p+q-1}}## ?
but my question was how should you make the transformation for an arbitrary probability density function?

The phrase "make the transformation" is ambiguous even if you specify the transformation of coordinates has the form ##x' = Ax + B## and that ##[a,b]## is transformed to ##[u,v]##.

Given a probability density ##f(x)## and a transformation of coordinates ##T(x) = Ax + B## such that ##T(a) = u ## and ##T(b) = v##, I think what you want is to define a probability density ##g(x)## satisfying the requirement that ##\int_{u_1}^{v_1} g(x) dx = \int_{T^{-1}(u_1)}^{T^{-1}(v_1)} f(x) dx ## for all intervals ##[u_1,v_1]##

The ##g(x)## satisfying that requirement is ##g(x) = (1/A) f(T^{-1}(x)) = (1/A) f(\frac{x-B}{A})##

In the integral ##\int_{T^{-1}(u_1)}^{T^{-1}(v_1)}(1/A) f(\frac{x-B}{A}) dx ## make the change of variables ##y = T^{-1}(x) = \frac{x-B}{A} ## , ##dy = \frac{dx}{A}##, ##dx = A\ dy##.

The integral is transformed to ##\int_a^b (1/A) f(y)\ A dy = \int_a^b f(y)dy = \int_a^b f(x) dx##

(If ##T## was a more complicated transformation the choice ##g(x) = (1/A) f(T^{-1}(x)## might not work since a substitution might not produce the simple relation ##dx = A\ dy##)

The conditions:
##T(a) = u = Aa + B##
##T(b) = v = Ab + B##
imply
##A = \frac{v - u}{b-a}##
##B = u - Aa = u - \frac{v-u}{b-a}(a) ##

Last edited:

## 1. How can I change the support of a probability density function?

To change the support of a probability density function, you can use a transformation function. This function maps the original support to a new support, allowing you to change the range of values for your probability density function.

## 2. What is the purpose of changing the support of a probability density function?

Changing the support of a probability density function can help in adjusting the distribution to better fit the data or to make calculations easier. It can also be used to create new distributions by transforming existing ones.

## 3. Can I change the support of any probability density function?

Yes, you can change the support of any probability density function as long as it is continuous and has a defined support. This includes commonly used distributions such as the normal, uniform, and exponential distributions.

## 4. How do I choose the transformation function for changing the support of a probability density function?

The choice of transformation function depends on the desired change in support and the properties of the original distribution. Commonly used transformation functions include logarithmic, exponential, and power functions.

## 5. Are there any limitations to changing the support of a probability density function?

While changing the support of a probability density function can be useful, it may also alter the shape and properties of the original distribution. It is important to carefully consider the implications of changing the support and to assess the resulting distribution to ensure it is still appropriate for the data.

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