Problem with a factor in a convolution of two probability density functions

Question

In a problem I'm working on I have analytical expression of two probability density functions $p_1(v_1),p_2(v_2)$ of two variables $v_1$ and $v_2$, which represent velocities of two particles.

I wish to find the probability density function of the center of mass velocity, which is $v=\frac{v_1+v_2}{2}$

To obtain the pdf of $v$ I'm using the convolution formula $q(v)=\int^{\infty}_{-\infty}p_1(v)p_2(v-\tau)d\tau$

My problem is that by using the convolution formula I obtain $q(v_1+v_2)$ rather than $q(\frac{v_1+v_2}{2})$

Can some one please explain where the factor 2 needs to be taken in account in the convolution formula, and provide a proper mathematical explanation.

Answer 1

If $x \mapsto f(x)$ is the density of $v_1 + v_2$, then the densité of $\frac{v_1+v_2}{2}$ is $x\mapsto 2 f(2x)$.

In general, if you have the PDF of $X$: $f$, the PDF of $\varphi(X)$ is $x \mapsto f(\varphi^{-1}(x)) {\varphi^{-1}}'(x)$ if $\varphi$ is bijective and its inverse is differentiable (to prove it, recall that the PDF is the derivative of cumulative distribution functions).