I have a question which is as follows:
The random variable U has the pdf $n \dfrac{u^{n-1}}{\theta^{n}}$ , $ 0 \le u \le \theta$
This is not independent of the parameter θ. Let Y=U/θ. Use the Jacobians to find the pdf of y?
I know that the answer to the pdf should be $n {y^{n-1}}$, $ 0 \le y \le 1$,
but I don't know how to derive it.
Can someone please enlighten me with some thoughts or steps for this. I know how to take the Jacobian but I don't know how to relate it to the problem. Thank you.
To find the pdf $f_Y(y)$ of a function $y=g(x)$ of a random variable with a pdf $f_X(x)$ you can use the identity: $$f_Y(y)=f_X(g^{-1}(y))\bigg|\frac{\mathrm d g^{-1}(y)}{\mathrm dy}\bigg|$$ Here you have $f_X(x)=n \dfrac{x^{n-1}}{\theta^{n}} $ and $g(x)=\frac{x}{\theta}$.
So $g^{-1}(y)=y \theta$ and $\bigg|\frac{\mathrm d g^{-1}(y)}{\mathrm dy}\bigg|=\theta \ $ (your Jacobian).
Then $f_X(x)=n \dfrac{x^{n-1}}{\theta^{n}}=\frac{n}{\theta}\left(\dfrac{x}{\theta}\right)^{n-1}\ $ and $ \ f_X(g^{-1}(y))=\frac{n}{\theta}y^{n-1}$. Putting it all together yields the result.