I know from that Pareto's pdf is: $$f({\bf y};\theta,x_m) = \theta x_m^{\theta} y^{-(\theta+1)} \mathbf{I}_{x_m<y}$$
Now the joint density is $$f({\bf y_1,\bf y_2,\dots,\bf y_n};\theta,x_m) = \theta x_m^{\theta} y_1^{-(\theta+1)}\theta x_m^{\theta} y_2^{-(\theta+1)},\dots,\theta x_m^{\theta} y_n^{-(\theta+1)}= \theta^n x_m^{\theta n}\prod_{i=1}^{n} y_i^{-(\theta+1)}\mathbf{I}_{x_m< \min(y_1,y_2,\dots,y_n)}$$
But in another post on Stack Math I found another form of the joint density which you can found it here, and the original poster writes it as :
$$f_n({\bf x}; \theta) = a^n b^{-n} \prod^n_{i=1}\left( \frac{b}{x_i} \right) ^{1 + a} $$
or in my notation : $$f_n({\bf y}; \theta) = \theta^n x_m^{-n} \prod^n_{i=1}\left( \frac{x_m}{y_i} \right) ^{1 + \theta} $$
while my joint density is different:
$$ f_n({\bf y}; \theta) = \theta^n x_m^{\theta n}\prod_{i=1}^{n} y_i^{-(\theta+1)}\mathbf{I}_{x_m< \min(y_1,y_2,\dots,y_n)}$$
What modifications has she/he made in order to write it in that way ?
Can someone explain me specifically step by step how she/he has done it ? Thank you very much in advance.
$$f(y;\theta,X_m)=\theta X_m^{\theta} \frac{1}{y^{\theta+1}}$$ \begin{align*} f(y_1,y_2,\dots,y_n;\theta,X_m) &= \theta X_m^{\theta} \frac{1}{y_1^{\theta+1}}\theta X_m^{\theta} \frac{1}{y_2^{\theta+1}},\dots,\theta X_m^{\theta} \frac{1}{y_n^{\theta+1}} \\ &= \theta^{n} X_m^{\theta n} \prod_{i=1}^{n} \frac{1}{y_i^{\theta+1}}\\ &= \frac{\theta^{n} X_m^{\theta n}}{X_m} \prod_{i=1}^{n} \frac{X_m}{y_i^{\theta+1}}\\ &= \theta^{n} \left(\frac{1}{X_m}\right)^{n} \prod_{i=1}^{n} \frac{X_m X_m^{\theta }}{y_i^{\theta+1}}\\ &= \theta^{n} X_m^{-n} \prod_{i=1}^{n} \frac{X_m^{\theta +1}}{y_i^{\theta+1}}\\ &= \theta^{n} X_m^{-n} \prod_{i=1}^{n} \left(\frac{X_m}{y_i}\right)^{\theta+1} \end{align*}