If $f(x)$ belong to set of probability distributions, then what can be deduced about $\frac{1}{a} f(\frac{1}{a}\cdot x)$ My question is in the context of probability distributions, whose Fourier transforms (characteristic function) almost always exit.
If $f(x)$ be some function such that $ \int_{-\infty}^\infty f(x) \, dx=1,$ 
then what can be said/deduce about $ \frac{1}{a} \int_{-\infty}^\infty f\left(\frac{1}{a}x\right) \, dx,$ where $a \in \mathbb{R^+}?$

Hint: use change of variable $z=\frac{x}{a}$.
