Question

Suppose that X has the beta distribution with parameters α and β. Show that 1 − X has the beta distribution with parameters β and α.

Short answer

1 − X has the beta distribution with parameters β and α.

Step-by-step solution

Given information

X has the beta distribution with parameters α and β. We need to prove that 1 − X has the beta distribution with parameters β and α.

Proof of 1 − X has the beta distribution with parameters β and α.

If X has beta distribution then

\(f\left( x \right) = \frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. \left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }}{x^{\alpha - 1}}{\left( {1 - x} \right)^{\beta - 1}}\)

Now replacing x by 1-x we get

\(\begin{aligned}{}f\left( {1 - x} \right)& = \frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. \left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }}{\left( {1 - x} \right)^{\alpha - 1}}{\left( {1 - \left( {1 - x} \right)} \right)^{\beta - 1}}\\ &= \frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, {\left( \alpha \right)} \,}} \right. \left| \!{\overline {\, {\left( \beta \right)} \,}} \right. }}{\left( {1 - \left( {1 - x} \right)} \right)^{\beta - 1}}{\left( {1 - x} \right)^{\alpha 1}}\end{aligned}\)

Thus it is proved that 1 − X has the beta distribution with parameters β and α.