Probability density function in Rayleigh distribution

Question

It says that

$$ f(x;\theta) = (x/\theta)e^{-x^2/(2\theta^2)}, x>0 $$

is the Rayleigh distribution.

And asks to verify that $f(x;\theta)$ is a legitimate pdf.

Can you explain how to verify legitimate pdf and what $f(x;\theta)$ means?

Thank you.

Answer 1

What are the properties that a distribution satisfies?

In particular, does the density integrate to $1$ over $(0,\infty)$? Verify that it does.

Also the notation $f(x;\theta)$ is represents the density, where $x$ is the variable, and $\theta>0$ is a fixed parameter.