I have searched for quite a bit on the Internet and I wonder whether I can double check with you my concept about the joint p.d.f. in terms of the n variables i.i.d to a Beta distribution.
For example, I let the p.d.f. of X as
$f_x(x)=\theta x^{(\theta-1)}, 0<x<1$
Can I write down the joint p.d.f. of (X1,X2,...,Xn) as $f_x(x1,x2,...,x_n)=\theta^n(x_1x_2...x_n)^{\theta-1}$ ?
I am confused about the joint p.d.f. concept if I am only given the p.d.f. of X.
Thank you very much for reading and any help!
Yes, the joint PDF of independent variables is the product of the PDFs of the individual variables. Note, independence is crucial here.
So we have $$ f_{X_1,\ldots, X_n}(x_1,\ldots x_n) = f_{X_1}(x_1)f_{X_2}(x_2)\ldots f_{X_n}(x_n).$$ Since they are also identically distributed, with the distribution $f_X(x)=\theta x^{\theta-1},$ we have $$ f_{X_1,\ldots, X_n}(x_1,\ldots, x_n) = \theta x_1^{\theta-1}\theta x_2^{\theta-1}\ldots \theta x_n^{\theta-1} = \theta^n (x_1x_2\ldots x_n)^{\theta-1}$$