# dirac delta – Derivative of the probability transformation formula

Let $$f: mathbb {R} ^ n rightarrow mathbb {R}$$ to be a measurable function of Borel. (It can further be assumed that $$| frac { partial f} { partial z_i} |> 0$$.

$$overrightarrow {Z} = (Z_1, Z_2, points, Z_n$$ is a random variable vector with probability density function $$p _ { overrightarrow {Z}}$$. Let $$Y: = f ( overrightarrow {Z})$$ be a random variable with probability density function $$p_ {Y}$$.

By the probability transformation formula, we have

$$p_Y (y) = int _ { mathbb {R} ^ n} p _ { overrightarrow {Z}} ( overrightarrow {z}) delta (yf ( overrightarrow {z}) d overrightarrow {z }$$.

We want to know $$frac {dp_Y} {dy}$$? I do not really know how we take the derivative to the $$delta (y-f ( overrightarrow {z})$$ as $$y$$ also depends on $$overrightarrow {z}$$.

Moreover, we can assume that $$Z_i$$ is i.i.d, so we could also have
$$p_Y (y) = int _ { mathbb {R}} p_z (z_n) dz_n dots int _ { mathbb {R}} p_z (z_1) delta (yf ( overrightarrow {z}) dz_1$$.

Someone has any idea?

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