Continuous random variables $A$, $B$, $X$ and $Y$ are independent. $A$ and $B$ have the same distribution. $X$ and $Y$ have the same distribution. What is the expected value of $(X-Y)\operatorname{sgn}...

I want to compute the expectations for the joint density$$f(x, y)= \begin{cases}12 y^2 & 0 \leq y \leq x \leq 1, \\ 0 & \text { otherwise } .\end{cases}$$Here the underlying probability ...

Let $X,Y$ be two i.i.d.I am trying to bound the expectation of how afar from one another they can get?That is, $E[|X-Y|]$. I know that:$$ E[X-Y] = E[X]- E[Y] = 0$$But what about $|X-Y|$?One ...

Let $X \in L^2$. Then $Z = E[X|G]$, for some sub $\sigma$-algebra $G$, is the orthogonal projection of $X$ onto $L^2(G)$. That is $Z$ is the random variable such that for every $G' \in G$:$$\int_{G'} ...

We flip a coin $n$ times. The probability of tails is $p$.Let $X$ be the number of occurrences of a run of heads followed by a longer run of tails. We can also agree that if the sequence begins with ...

A few years prior, an acquaintance of mine tackled a problem inspired by something in our statistics class, which basically was the idea of "what is the expected distance from the starting point ...

I have a problem related to the idle game Clicker Heroes, in finding the expected $\log_{10}$ value of the rewards gained from defeating bosses throughout an ascension. For boss number $k$, the reward ...

On a unit disk, how should a string of length $2$ be laid in order to minimize $E$, the expected shortest distance between a uniformly random point on the disk and the string?For example, the ...

Let $\varepsilon_1, \dots, \varepsilon_n$ be independent random variables with $E(\varepsilon_i) = 0$.Let $f: [0,1] \to \mathbb{R}$ be a Lipschitz function with constant $K > 0$, i.e.,$$|f(x) - f(...

I need help understanding law of iterated expectation in its most basic form. I want to make sure my understanding is correct.We have $E(E(X|Y)) = E(X)$.We know $E(X) = \sum xP(X=x)$ for discrete ...

Assume I have a random variable $X$ with distribution function $F$.Its expectation would be the Stieltjes integral wrt the distribution function:$$E[X]=\int_{-\infty}^{\infty} t d F(t)$$Assume now ...

A decision must be made as to whether to do nothing and accept a cost of X or eliminate the cost by enaging a decision that costs nominal amount Y, where Y is approximately 1/3 of X.Without getting ...

Starting with a set of $n$ dice, roll all of them together, and remove any that come up 1, 2, or 3. Repeat this until all the dice are removed.However, if at any point you have one die left, you can ...

Let $W \subseteq \mathbb{R}^2$ be a finite set of vectors, $ P$ be a probability distribution on $W$, and $V_0\in \mathbb{R}^2$ (for simplicity it suffices to consider $V_0$ where both coordinates are ...

You roll 100 dice onto a table. You then repeatedly remove sets of 6 distinct dice (i.e. the set 1, 2, 3, 4, 5, and 6) until it is no longer possible to do so. What is the expected value of the sum of ...