Is there a clever way to find two density functions, $f$ and $g$, that satisfy the following conditions?
$\begin{align*} \int_{\infty}^{m}\int_{-\infty}^{\infty}f(w)f(w+z)\,dw\,dz&=\int_{\infty}^{m}\int_{-\infty}^{\infty}g(w)g(w+z)\,dw\,dz\\ \int_{\infty}^{m+2}\int_{-\infty}^{\infty}f(w)f(w+z)\,dw\,dz&=\int_{\infty}^{m+1}\int_{-\infty}^{\infty}g(w)g(w+z)\,dw\,dz\\ \int_{-\infty}^{\infty}f(w)\,dw&=\int_{-\infty}^{\infty}g(w)\,dw=M\\ \end{align*}$ where $f\gt 0$ and $g\gt 0$ almost everywhere?
for $m\in (-\delta,\delta)$ and $\delta$ is some small number.
My main intent is to come up with two i.i.d. random variable, X' and X'' and $Y$ and Y'', such that \operatorname{\mathbb{Pr}}(m> Y'-Y'')=\operatorname{\mathbb{Pr}}(m>X'-X'') for $m \in (-b,b)$ for some $b$ small enough, while \operatorname{\mathbb{Pr}}(m+2> Y'-Y'')=\operatorname{\mathbb{Pr}}(m+1> X'-X'').Is this possible?
Thanks so much in advance for your much appreciated help.