Moments of beta distribution

Question

I have a random variable given by $$ Y = a \cdot X \;, $$ where $X$ follows a beta distribution and $a$ is a simple constant. I want to find the moments of $Y$.

I am aware of the general formula for moments of the beta distribution and want to specifically know how this constant $a$ affects this formula!

Thank you in advance, Sam

Answer 1

For any distribution, the moments are given by

$$\mu_{X,n}=\int_{\mathbb R}x^n\text{ pdf}_X(x)\,dx.$$

Then replacing $x$ by $y=ax$,

$$\mu_{Y,n}=\int_{\mathbb R}y^n\text{ pdf}_Y(y)\,dy=\int_{\mathbb R}(ax)^n\text{ pdf}_X(x)\,dx=a^n\mu_{X,n}.$$

Answer 2

You have a random variable $X$ with values in the interval $[0, 1]$. Let's say $f_X : [0, 1] \to [0, \infty)$ is its pdf.

The scaled r.v. $Y := a X$ has values in $[0, a]$. Its pdf is $f_Y(y) = a^{-1} f_X(y/a)$. The scaling keeps the total mass equal to one.

The $n$-th moment of $Y$ is $$ M_Y^{(n)} = \int_0^a y^n f_Y(y) dy = \int_0^1 (a x)^n a^{-1} f_X(x) d(a x) = a^n M_X^{(n)} . $$