Suppose we have a non-stationary joint probability density function (PDF):
$$ p(x, y, t) = p_x(x,t)p_y(y,t) \quad \text{s.t.} \quad \int_{\mathcal{X} \times \mathcal{Y}} p(x,y,t) \mathrm{d}(x, y) = 1. $$
Suppose that $p_y$ is defined by a non-stationary dirac-delta distribution that follows a bijective function $f$, i.e.:
$$ p_y(y, t) = \delta(y - f(t)) \quad \text{s.t.} \quad f(f^{-1}(y)) = y, \quad f^{-1}(f(t)) = t. $$
If we apply a change of variables to remove $t$, we get
$$ p'(x, y) := p(x, y, f^{-1}(y)) = p(x, f^{-1}(y))\delta(y - y) = p(x, f^{-1}(y))\delta(0). $$
How can $p'(x, y)$ be interpreted?