Let $X_{1},X_{2},\ldots ,X_{n}$ be a random sample of size $n$ from a population distribution $F$. I want to find the following:
1. the joint P.d.f of $X_{1},X_{2},\ldots ,X_{n}$.
2. the marginal probability distribution of $X_{j}$ for any $j$ in $1\leq j \leq n$.
This is my attempt for 1.
Given $F$, the joint pdf of the random sample is
\begin{align*} F\left(X_{1},X_{2},\ldots ,X_{n}\right) & =P \left(X_{1}\leq x_1, X_{2}\leq x_2, \ldots X_n\leq x_n \right)\\ &=P(X_1 \leq x_1)P(X_2\leq x_2)\dot{} \ldots \dot{}P(X_n\leq x_n) \\ &=\left[P(X_1\leq x_1)\right ]^n \qquad \because X_j\text{'s are identical} \end{align*}
Here are my questions: First, iIs my attempt for 1. right. Is there a better way of doing it.Second, I would like some help with 2. I know the $X_j$'s will all have the same marginal distributions, but I don't know how to justify it.
Thanks.