Problem 1.

There is a frog on an integer line, initially sitting at a point 0. Each minute, it jumps left with probability $p$ and right with probability $1-p$. Let $T_n$ be a random variable representing the position of the frog after $n$ minutes pass (i.e $n$ jumps happen), what is the $\mathbb{E}[T_n]$ and what is $\text{Var}(T_n)$?
(Remark for knowledgeable people: Yes, this is a question about finding the variance of a random walk and yes, it was asked during an interview)

Problem 2.

Two fair dice are rolled, the number that comes up on one of them is $D_1$, the number that comes up on another one is $D_2$. Let $X = \text{min}(D_1, D_2)$, and let $Y = \text{max}(D_1, D_2)$. Find the correlation between $X$ and $Y$.

Problem 3.

Recall what is a Poisson Distribution, and find its variance. Same with the Geometric distribution.
(Once you make sure you got the distributions correctly, try to do it as if it was an interview: i.e without looking into your notes, googling stuff, etc...)

Problem 4.

Suppose $n$ distinguishable balls are randomly distributed into $r$ boxes. If $S_r$ is the number of empty boxes, find the $\mathbb{E}[S_r]$ as well as $\text{Var}(S_r)$.

Problem 5.

If there is a 50\% probability that bond A will default next year and a 30\% probability that bond B will default. What is the range of probability that at least one bond defaults and what is the range of their correlation?
(Remark-Hint: You should not be scared of the words "bond" and "default". It is fine if you do not know their meaning, you can google those. From the mathematical standpoint "bond defaulting" is simply "an event". Thus, basically this problem is about two events, A and B, their possible correlation and its influence on the the $\mathbb{P}(A \text{ or } B).$)

Problem 6.

Let $X$ and $Y$ be non-constant two random variables, for which both mathematical expectation as well as the variance exist. Find the values of constanst $c$ and $d$ (in terms of $X$ and $Y$) that minimise \[ \mathbb{E}[ \, (Y - cX - d)^2 \, ] \] Compare these values to the formulas for $a$ and $b$ from the last exercise from the classwork, comment on your result.