I am trying to derive the classical probability density function to find the harmonic oscillator at position $x$. I am confused between the random variables involved here $x, t$ and not able to proceed. $$P_X(x) = \int{dx'\delta(x' -x) } $$

How do I make change of variable to $x'=A\sin\omega t$ in the above expression.

I understand how to find the probability density function using $dt=\frac{dx}{v(x)}$ and then using total energy is a constant. I am trying to arrive at the same result summing the $\delta$ functions.