Problem 1.

Let $U_1$ and $U_2$ be two independent uniform random variables drawn from $[0,1]$. Let $A = \text{min}(U_1,U_2)$ and $B = \text{max}(U_1,U_2)$. What’s the correlation between random variables $A$ and $B$?

Problem 2.

a) Recall or prove Markov's inequality: Let $X$ be a non-negative random variable. Then for any $a \geq 0$ it is true that $\mathbb{P}(X \geq a) \leq \mathbb{E}[X]/a$;
b) Recall or prove Chebyshev's inequality: Let $X$ be a random variable whose mathematical expectation exists. Then for all $a \geq 0$, it is true that $\mathbb{P}(|X - \mathbb{E}[X] \geq a) \leq \text{Var}(X) / a^2$;
c) Let $\lambda$ be constant, and let $G$ be a random graph with $n$ vertices, where any edge is drawn with probability $p = \lambda \frac{\log n}{n}$ independent of other edges. Show that if $\lambda < 1$ then as $n \to \infty$, the probability that $G$ is connected is tending to 1. Also show that if $\lambda > 1$, then as $n \to \infty$ the probability that $G$ is connected is going to 0.