Graph problem with expectation

Consider an undirected graph G = (V, E) representing the social network of friendship/trust between employees. We partition the set of node triples into four sets T0, T1, T2, and T3. A node triple v1, v2, v3 belongs to T0 iff no edge exists between the nodes v1, v2, and v3, – T1 iff exactly one of the edges (v1, v2), (v2, v3), and (v3; v1) exists,- T2 iff exactly two of the edges (v1, v2), (v2, v3), and (v3, v1) exist,- T3 iff all of the edges (v1, v2), (v2, v3), and (v3; v1) exist. |T3| denotes the number of connected triples in the graph that is the quantity we need to estimate. if i apply the following algorithm:

• Sample an edge e = (a, b) uniformly chosen from E

• Choose a node v uniformly from V \ {a, b}

• if (a, v) ∈ E and (b, v) ∈ E then x = 1, else x = 0

How can I show that |T1| + 2|T2| + 3|T3| = |E|(|V | − 2)?

Find expectation with Chernoff bound

We have a group of employees and their company will assign a prize to as many employees as possible by finding the ones probably better than the rest. The company assigned the same $$2$$ tasks to every employee and scored their results with $$2$$ values $$x, y$$ both in $$[0, 1]$$. The company selects the best employees among the others, if there is no other employee with a better score in both tasks.

Knowing that both scores are uniformly distributed in $$[0, 1]$$, how can i proof that the number of the employees receiving the price is estimated near to $$\log n$$, with $$n$$ the number of the employees, having high probability?

I need to use Chernoff bound to bound the probability, that the number of winning employees is higher than $$\log n$$.

Find expectation and calculate Chernoff bound

We have a group of employees and their company will assign a prize to as many employees as possible by finding the ones probably better than the rest. The company assigned the same 2 tasks to every employee and scored their results with 2 values x,y both in [0,1]. The company selects the best employees among the others, if there is no other employee with a better score in both tasks.

Knowing that both scores are uniformly distributed in [0,1], how can i proof that the number of the employees receiving the price is estimated near to logn, with n the number of the employees, having high probability?

I need to use Chernoff bound to bound the probability, that the number of winning employees is higher than logn.

What magic item benefits are built into the expectation of the game?

I often read that magic item progression is built into the game’s difficulty, and that it’s important for PCs to keep up with the progression in order to not fall behind.

Exactly what progression is “built in” and “necessary” for a PC?

For example, I usually buy an anklet of translocation for most of my PCs around level 5, and I also usually buy a ability-boosting item around level 6. I imagine the game anticipates that ability boost, but it probably doesn’t anticipate the anklet.

Similarly, most players I know like to buy a handy haversack at some point between levels 6 and 12. I don’t imagine the game designers built that into the CR of the monsters.

The Book of Exalted Deeds, on pg 31, provides a chart of progression for characters who take a vow of poverty. Is this about what the game designers had in mind for magic item progression? (And if so, how do you account for the bonus feats?)

Another way of asking this same question is this: there is a published Wealth by Level chart. How much of that gold is sort-of allocated to necessary magic items (resistances, weapon enhancements, ability boosts, etc.) and how much of it is allocated to strange wondrous items don’t directly influence the PC’s numeric values? (like an anklet of translocation, a wand of silent image, a handy haversack, etc.)

What is the correct user expectation on text alignment, when English is used together with Arabic language?

I was wondering, what is the correct user expectation, when English is being used together with Arabic, within a sentence.

For instance, in the 1st line, the text is aligned toward right. How I type are

1. I type “hello”
2. I type [space]
3. I type انا احبك

In the 2nd line, the text is aligned toward left. How I type are

1. I type انا احبك
2. I type [space]
3. I type “hello”

I was wondering, which text alignment is the correct user expectation?

p/s I have 0 background in Arabic language. The only thing I know, is Arabic writing system is from right to left.

Ratio of expectation involving random unit vectors

Let $$u=(u_1,…,u_n), v=(v_1,…,v_n)$$ be two random vectors independently and uniformly distributed on the unit sphere in $$\mathbb{R}^n$$. Define two other random variables $$X=\sqrt{\sum_{i=1}^nu_i^2v_i^2}$$, $$Y=|u_1v_1|$$. Consider the following ratio of expectation: $$r_n(\alpha)=\frac{\mathbb{E}\{\exp[-\frac{\alpha^2-\alpha^2X^2+\alpha X}{2}]\}}{\mathbb{E}\{\exp[-(\alpha^2-\alpha^2Y^2+\alpha Y)]\}}$$ Does there exist a finite upper bound for $$r_n(\alpha)$$, independent of $$\alpha$$, for all $$\alpha\geq0$$?

Ratio of expectation involving random unit vectors

Let $$u=(u_1,…,u_n), v=(v_1,…,v_n)$$ be two random vectors independently and uniformly distributed on the unit sphere in $$\mathbb{R}^n$$. Define two other random variables $$X=\sqrt{\sum_{i=1}^nu_i^2v_i^2}$$, $$Y=|u_1v_1|$$. Consider the following ratio of expectation: $$r_n(\alpha)=\frac{\mathbb{E}\{\exp[-\frac{\alpha^2-\alpha^2X^2+\alpha X}{2}]\}}{\mathbb{E}\{\exp[-(\alpha^2-\alpha^2Y^2+\alpha Y)]\}}$$ Does there exist a finite upper bound for $$r_n(\alpha)$$, independent of $$\alpha$$, for all $$\alpha\geq0$$?

Root of the expectation of a random rational function

I am trying to figure out a formula for the unique $$\lambda>1$$ such that $$\mathbb{E}\bigg[\frac{X}{\lambda -X}\bigg]=1$$ where $$X$$ is a discrete random variable taking values in $$\{\frac{1}{n},…,\frac{n-1}{n},1\}$$, distributed w.r.t. some distribution $$\mu$$.

We can rewrite the expression above which yields $$\sum_{k=1}^n \frac{\mu(\frac{k}{n})\frac{k}{n}}{x-\frac{k}{n}} = 1.$$

I know that there are no closed solutions for the roots of such a function, since they are based on solving for zeros of a high degree polynomial. Still, I think I miss some obvious results here how to analyse such a function.

I’d appreciate any kind of help. Thank’s a lot!

Calculating expectation and variance for having rolled 1 and 6 twice out of rolling a die 12 times

First i have calculated the probability to get each possible number {1,2,3,4,5,6} twice from 12 rolls(A).

$$Pr[A]=\frac{\binom{12}{2,2,2,2,2,2}}{6^{12}}.$$

Then there are 2 random variables:

X-number of times that 1 was received.

Y-number of times that 6 was received.

Before calculating $$E(X),Var(X),E(Y),Var(Y)$$ i’m uncertain of how i should calculate the probabilities of X and Y

Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality

In my work I wish to obtain a lower bound for the term below, independent of the vector $$x$$. Here the expectation is taken over $$h$$, a standard random Gaussian vector of length $$n$$. The vector $$x$$ is fixed. The minimum is taken over all $$\{i_1,\dots,i_L\} \in \{1,\dots,n\}$$. Can this be done using the Sudakov-Fernique inequality? $$\mathbb{E}_{h} \min _{i_{1}, \ldots, i_{L}}\left[\sum_{j\neq i_1,\dots,i_L}h_j\mathrm{sign}(x_j^*)\right].$$