Interchange of expectation and limit
$E\left[ S_{t} \right] = E\left[ \lim\limits_{n \to \infty}\sum_{i=1}^{n} R_{i}(t) \right]=\lim\limits_{n \to \infty}\sum_{i=1}^{n}E\left[ R_{i}(t) \right]=0$Each $R_{i}$ is an independent coin toss...
View ArticleExpected value of squared random variable if expected value of random...
Let $E[X]=0$ them How is Expected value of $E[X^2]$ ? in my opinion is $0$ But I need confirmation and explanationIn general, if $ (\Omega,\Sigma,P) $ is a probability space and $ X: (\Omega,\Sigma)...
View ArticleBasic Die Game expected payout after re-roll
Alice rolls a fair 6−sided die with the values 1−6 on the sides. She sees that value showing up and then is allowed to decide whether or not she wants to roll again. Each re-roll costs$1. Whenever she...
View ArticleIf $X_1,\dots,X_n$ are independent and identically distributed and $S_n$ is...
Consider the following exerciseIf $X_1,\dots,X_n$ are independent with same distribution and $S_n=X_1+\cdots+X_n$, prove that$$\mathbb{E}[X_i \mid S_n]=\frac{S_n}{n}$$Proof: Since the variables have...
View ArticleExpected score on an increasingly multiple choice test
If a multiple-choice test (where all questions are worth the same) has $n$ choices on each question, random guessing gets you a score of $\frac1n$ on average. But what if the number of choices...
View ArticleIs the optimal strategy to find the special coin the greedy strategy
Problem. There's $n$ coins, $n-1$ are fair and one always shows heads, but all coins look the same. You are allowed to flip $m$ coins, where $m < n$ and the choice of the $i$th flip can depend on...
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Intuition behind expected/mean time to absorption formula for Markov Chains
No matter how much I try I cannot understand the intuition behind this formula. I do not understand why there is a $1$ to begin with, and I do not understand why you multiply all probabilities by $u$...
View ArticleExpectation and variance of bivariate skew normal distribution
I am fitting a bivariate skew normal distribution to a 2D data through the sn package in R. I get a $2 \times 1$ vector of location and shape parameters $\beta$ and $\alpha$ as well as a $2 \times 2$...
View ArticleExpected Length of Walk on Truncated Icosahedron
Consider a truncated icosahedron with 12 pentagons and 20 hexagons. Starting from a hexagonal face, we go to any neighboring polygon randomly with equal probability. What is the expected number of...
View ArticleExpected number of draws to sample an element using weighted sampling without...
Let us have $k$ elements of weights $w_1,\ldots,w_k$ where the sum of the weights $w_1+w_2+\ldots+w_k=k$, where $w_i\geq 0$ for all $i$. We sample elements using weighted sampling (probabilities...
View ArticleUpper bound on expectation given probability [closed]
Let $X$ be a positive random variable. I know that $\mathbb{P}[X \leq a] \geq q$. Any hints about how to find an upper bound on $E[X]$ in terms of $q$ and $a$?Using Markov inequality, I can obtain...
View ArticleCorrectness of notation $\mathbb{E}_{x \sim p(x), y \sim p(y|x)} [f(x, y)]$
Let x be a random variable sampled from distribution $p(x)$. Let $y$ be a random variable sampled from conditional distribution $p(y|x)$.I'd like to express$ \int_{x} \int_{y} f(x, y) p(x)p(y|x)dxdy$...
View ArticleNon-decreasing expectated value of non-decreasing function with family of...
I am trying to prove the Lemma(3.4.2)(1) of book "Testing statistical hypotheses" by Lehmann and Romano, its statement is " If $\{p_{\theta}(x)\}$ be a family of densities on the real line with...
View ArticleFactoring integral term on joint expectation
I am reading a paper titled Transformer can do Bayesian inference.I am lost in proof of insight 1, in which they derive (equation 3):$$-\int_{D,x,y}p(x,y,D)\log q_{\theta}(y|x,D) =...
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$\int_0^{\pi/2}\int_0^{\pi/2}\frac{(\tan\alpha)(\tan\beta)}{\tan\alpha+\tan\b...
In the diagram, $\alpha$ and $\beta$ are independent uniformly random real numbers in $\left(0,\frac{\pi}{2}\right)$.What is $\mathbb{E}(h)$?Superimposing a cartesian coordinate system, the equations...
View ArticleExpected winning time in fair gambler's ruin problem using martingale
In fair gambler's ruin problem, we already knew that the expected time of winning is $$E(\tau_n|\tau_n<\tau_0)=\frac{n^2-k^2}{3}$$ where $k$ is how much money we have in the beginning and $\tau_i$...
View ArticleExpectations of mean squared error
I am currently studying the expected mean squared error and the derivations of this are as follows:\begin{align}&E[(y_i - \hat{y}_0 | x_0]\\&=E [ (f(x) + \epsilon_i - \hat{f}(x_i))^2 |...
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Understand the meaning of $n_j$ in expectation problem
This is Self Test Exercise 7.8 of Sheldon's A First Course in Probability. I'm lost at what the question is asking, and illustrate this with an example.Suppose $r = 10$. That means that there are $10$...
View ArticleInequalities regarding CDF, expected value
Let F be the distribution function of a random variable $X$ such that $\mathbb E[X]=10$. It follows thata) if $F(-10)=0$ then $P(X\geq 30)\leq 1/2$b) if $F(-10)=0$ then $P(X\geq30)\leq 1/3$c) if...
View ArticleFinding smallest and largest possible value of $E[XY]$
Let $P(X=1)=P(X=2)=P(Y=1)=P(Y=2)=\frac{1}{2}$.Find the smallest and largest possible value of $E[XY]$.My solution:If $X$, $Y$ are independent, then $E[XY] = (\frac{3}{2})^2 = \frac{9}{4}$.Now, let's...
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