Independence

I am selling my house, and have decided to accept the first offer exceeding $£ K$. Assuming that offers are independent random variables with common distribution function $F$, find the expected number of offers received before I sell the house.

Let $X$ and $Y$ be independent random variables with common distribution function $F$ and density function $f$. Show that $V = \max \left\{ X, Y\right\}$ has distribution function $\mathbb {P}\left(V \leq x\right) = F(x)^{2}$ and density function $f_{V}(x) = 2 f(x) F(x), x \in \mathbb {R}$. Find the density function of $U = \min \left\{ X, Y\right\}$.

The annual rainfall figures in Bandrika are independent identically distributed continuous random variables $\left\{ X_{r}: r \geq 1\right\}$. Find the probability that:

Let $\left\{ X_{r}: r \geq 1\right\}$ be independent and identically distributed with distribution function $F$ satisfying $F(y) < 1$ for all $y$, and let $Y(y) = \min \left\{ k: X_{k} > y\right\}$. Show that

Let $P_{i} = \left(X_{i}, Y_{i}\right), 1 \leq i \leq n$, be independent, uniformly distributed points in the unit square $[0,1]^{2}$. A point $P_{i}$ is called peripheral if, for all $r = 1,2, \ldots , n$, either $X_{r} \leq X_{i}$ or $Y_{r} \leq Y_{i}$, or both. Show that the mean number of peripheral points is $n\left(\frac{3}{4}\right)^{n-1}$.

Let $U$ and $V$ be independent and uniformly distributed on the interval $[0,1]$.