Expectation when a partial range is known
Let $X$ be a continuous random variable with a density $f_X(x)$ and let $A\subseteq \mathbb{R}$. Would it be true to claim that:$E[X | X\in A] = \int_{x\in A} x \cdot f_X(x) \cdot dx$?My understanding...
View ArticleCorrect expression of the loss function for a Negative Binomial distribution
ContextI am trying to obtain an expression for the loss function or expected "shortfall" when lead time demand is a discrete random variable that follows a Negative Binomial distribution. The context...
View ArticleIntuition for Dice Roll Expected Value problem
I'm studying a chapter on Expected Value right now, and there is an interesting formula here: Imagine a Bernoulli event where there a “success” (such as tossing a coin / rolling a 5 on a dice) occurs...
View ArticleExpected number of Pareto-optimal points
Suppose $S$ is a set of $n$ points in a plane.A point is called maximal (or Pareto-optimal) if no other point in $S$ is both above and to the right of that point.If each point in $S$ is chosen...
View ArticleExpected number of Pareto-optimal points for $d$-dimensional points
I was wondering if it is possible to generalize any of the solutions previous question for the 2-dimensional case to derive the size of the set of the Pareto optimal points for an arbitrary dimension...
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how does huber compute the $var(s_n)/E[s_n]$ and $var(d_n)/E[d_n]$?
How does Huber in book 'Robust statistical procedures' in chapter 1 compute the variance of certain statistical functions?He defines the mean square deviation to be $$s_n = \sqrt{\frac{1}{n} \sum...
View ArticleExpected number of correct guesses when identifying 5 people, blindfolded
The Game: You are blindfolded in front of a line of 5 people. Your task is to identify the ordering of the people in the line.Conditions of the Game:You know the five names of the people, but do not...
View Article$\mathbb E[X\mid Y]=Y$ and $\mathbb E[Y\mid X]=X$ implies $X=Y$
I'm sure this must've been asked before here but I couldn't find it. I'm trying exercise 11.9 from Le Gall's "Measure theory, Probability, and Stochastic processes". The objective is to prove what I...
View ArticleTurducken Hunt OR Optimal Shooting Strategy — Probability Problem with Moving...
Mordecai and Rigby are chasing down the Turkaden. Both starting at x=0, the Turducken 1 meter away. They start walking at the same rate towards x=1. Both only have one bullet in their guns and they can...
View ArticleMaximum of $E[|X+Z|^m]$ for $Z$ standard normal and $X$ independent of $Z$,...
Let $Z$ be a standard normal distribution. I am trying to find a solution to the following problem:\begin{align}&\max_{ x_1,x_2 \in \mathbb{R}, t\in[0,1]} (1-t) E[|x_1+Z|^m]+t...
View ArticleCards Shuffling
Imagine you possess20 cards, each numbered from1 to10, with each number appearing twice. You randomly select2 cards from the20 available cards. If they are identical in value, they are removed from the...
View ArticleDefining Random Variable, Expected Value for Number of Fixed Points given a...
Let $n \in \mathbb N$, and $\mathcal{K}$ be the set of permutations possible for a set $\{1,...,n\}$. Let $\sigma \in \mathcal{K}, $ such that $\sigma: [n] \to [n]$ is a randomly selected permutation....
View ArticleFormula for Laplace transform of the jump probability in a continuous time...
I am trying to understand the basics of continuous time random walks, and this formula has no explanation as far as I can find:$$\hat{p}_{n}(s) = \hat{\rho}(s)^n\cdot\frac{1-\hat{\rho}(s)}{s}$$Where...
View ArticleVerifying the inverse Laplace transform for a production-inventory problem:...
I am entirely self-taught when it comes to Laplace transforms, and I am seeking an independent opinion on my attempt to work out how to arrive at the below expression (note: I am interested...
View ArticleFinding the expected value of the minimum of an finite series of random...
Below is a problem I did. Part (a) is correct so you can skip over it. Part (b) is wrong. I want to know where I went wrong.Problem:Suppose that the length of time $Y$ that it takes a worker to...
View ArticleFisher information of the Rayleigh distribution
Problem description: Find the Fisher information of the Rayleigh distribution. I was satisfied with my solution until I saw that it disagreed with the solution obtained in one of the problem sets from...
View ArticleKing Flip Countdown
The card dealer deals from a set containing 10 regular decks of cards. This implies there are 40 cards of each rank and 520 cards in total. The dealer conducts a game where he thoroughly shuffles all...
View ArticleRoll Gamble Strategy
Q)Two fair 6-sided dice are thrown. Decide whether to retain the product of the shown numbers or roll both dice again for 4. What isthe expected value of your winnings using an optimal strategy?My...
View ArticleGame theory expected value
We play a game involving two players. Each player calls a number 1 or 2. If the sum of these numbers are odd (i.e. equal to 3), then player 1 gets 3 points and player 2 loses 3 points. If the sum of...
View ArticleProof of conditional Hoeffding lemma
I want to prove the following statement: For any random variable $X$ such that $a \leq X \leq b$ almost surely, for all $\lambda>0$ and a $\sigma$-algebra $\mathcal{G}$(let's say this is a...
View ArticleExpected number of steps to reach a state
Suppose I have a set of states $S_i, i\in \{0,1...n\} $ ,and $P(S_i \rightarrow S_j)$ be the probability of moving $S_i$ to $S_j$ in the next step, I was trying to find the expected number of steps to...
View ArticleProbability of empty bin, where the number of balls is based on another game...
Came across this interesting question:We flip a fair coin until we obtain our first heads. If the firstheads occurs on the kth flip, we are given k balls. We put them into 3bins labelled 1, 2, and 3 at...
View ArticleGame of toin coss. If you get TT, you repeat, what's the expected number of...
Suppose you have a game in which you stop if you get HH, HT or TH and you keep rolling if you get TT. You have to calculate expected number of tossesThe solution given in textbook (Heard on the street)...
View ArticleMonotonicity with expectation
I think the following is true but I cannot prove it.Let $Z_1, Z_2$ are two random variables defined on the same sample space $\Omega$. Suppose that $Z_1(\omega) < Z_2(\omega)$ for all $\omega\in...
View ArticleFair value of coin tossing game
A biased coin is flipped $100$ times. The head and tail probabilities are unknown and the probability of heads is uniformly distributed between $0\%$ and $100\%$. if you guess correctly, you get $\$1$;...
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$\int_0^{\pi/2}\int_0^{\pi/2}\frac{(\tan\alpha)(\tan\beta)}{\tan\alpha+\tan\b...
In the diagram, $\alpha$ and $\beta$ are independent uniformly random real numbers in $\left(0,\frac{\pi}{2}\right)$.What is $\mathbb{E}(h)$?Superimposing a cartesian coordinate system, the equations...
View ArticleTricky conditional expected value and random variable question
Consider two identically distributed, but not necessarily independent bivariate random variables $(O_{k},h_{k})$ and $(O_{l},h_{l})$, where $O_{k},O_{l} \in (0,m]$ and $h_{k},h_{l} \in \{0,1\}$. If for...
View ArticleThe Probabilistic Method, exercise 2.7.9
This is Exercise 2.7.9 from "The Probabilistic Method" by Noga Alon and Joel Spencer. It's stated as follow:Let $G = (V, E)$ be a bipartite graph with $n$ vertices and a list $S(v)$ of more than...
View ArticleExpected number of distinct objects in sampling with replacement
Given the set of numbers from 1 to n: { 1, 2, 3 .. n } We draw n numbers randomly (with uniform distribution) from this set (with replacement). What is the expected number of distinct values that we...
View ArticleCan $E(X)$ be defined using the Law of Large Numbers as "universal property"?
As an algebraist who occasionally teaches introductory statistics and probability, this question has been on my mind for a while:Can one use the Law of Large Numbers (LLN) to define the expectedvalue...
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