Method of Jacobians You need to use the method of Jacobians to find the joint density of $(\eta_1, \eta_2)$ and then use the joint density to conclude independence. Observe that due to independence of $\xi_1, \xi_2$ that their joint density function is given by $$f_{\xi_1, \xi_2}(x, y) = f_{\x...

Notice that $$\mathbb{P}(Y \leq y) = \mathbb{P}(\max(X_1, \dots, X_n) \leq y)\text{.}$$ Now, note that if the largest of $n$ numbers is less than $y$, it follows that each of the $n$ numbers must be less than $y$ as well. So, $$\mathbb{P}(\max(X_1, \dots, X_n) \leq y) = \mathbb{P}(X_1 \leq y \c...

Exercise: Let $L_n:=\{(x,\frac{x}{n})\mid0\leq x\leq1\}\subset\mathbb{R}^2$ for $n\in\mathbb{Z}_+$. Then consider $X=\{(1,0)\}\cup\bigcup_{n=1}^\infty L_n\subset\mathbb{R}^2$ together with the subspace topology. Is this topological space path-connected, connected, and/or locally connected? Explai...