What monotonicity of expected value implies [closed]
I've been dealing with this problem for a while but can't solve it.Consider a sequence of random variables $S = X_1, X_2,...$ which holds $E[X_{i-1}] < E[X_i]$ for all i.In addition, we denote by...
View ArticleName this measure of correlation
The covariance between two random variables $X$ and $Y$ can be expressed as a difference between two terms:$$\text{Cov}(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y]$$I am interested in the...
View ArticleExpected value of the log of a sum of half normal distributions
given $X_1, \ldots, X_n$ independent random variables following a centered gaussian distribution $\mathcal{N}(0, 1)$, I wish to find the expected value of $Y = \log(|X_1|+\ldots + |X_n|)$ (having the...
View ArticleDifferentiate the Normalization Term of the Binomial to Get the Expectation
I am working through the problems in Bishop's Pattern Recognition and Machine Learning; I am currently on exercise 2.4 The problem asks us to show that the mean of the binomial distribution by...
View ArticleGiven a random variable $Y$ follows $U(0,2\pi)$, what is the expected value...
Given a random variable $Y$ follows $U(0,2\pi)$, what is the expected value of $X = \sin(at+Y)$?
View ArticleEven moments are zero iff $X=0$?
Kind of a dumb question, but I wasn't able to find this in probability texts, either elementary (Larsen and Marx) or advanced (David Williams), so it's probably wrong.Is this false?Let $X$ be a random...
View ArticleLet $X_i\sim\text{Ber}(\frac{1}{2})$ be i.i.d. and $Y_i=\max(X_i,X_{i+1})$...
Problem: Let $X_1,X_2,\dots$ be independent random variables with $X_i\sim\text{Ber}(\frac{1}{2})$. Let $Y_i=\max(X_i,X_{i+1})$ and $Z_n=\sum_{i=1}^{n}Y_i$. Find $\text{E}(Z_n)$ and...
View ArticleQuestion about iterated expectation with a conditional outer expectation
I am trying to understand if I can simplify an expression. I feel like I can, but I am having a hard time breaking things down.Let $y,x$ be independent random variables with unknown distributions. Let...
View ArticleExpected number of items from resources with preservation chance and doubling...
BackgroundHi, I'm playing a game where you create items from resources and I would like to get some intuition on the expected number of items.For simplicity, I will assume that $1$ item is created from...
View ArticleExpected value of one-dimensional random walk with additive gain and...
Let's define a random walk starting at $S_{0}=0$ and$$S_{n+1} = \begin{cases} S_{n} + W & \text{with a probability of $p$}\\[0.25em] S_{n}L & \text{with a probability of $1-p$}...
View ArticleSimple Random Walk: expectation of number of times of hitting level k before...
Assume $S_0 = 0$ and let $N_k$ be the number of times of visiting level k before returning to origin. My textbook first claims that $P(N_k > 0) = \frac{1}{2k}$ and $P(N_k > j+1|N_k > j) =...
View ArticleCauchy-Schwarz for expected values of vector-valued functions
Given functions $f,g: \mathcal{X} \to \mathcal{H}$ lying in a vector-valued RKHS $\mathcal{H}_v$, i.e. outputting functions in the RKHS $\mathcal{H}$, what can we do about bounding the...
View ArticleConvergence of expectation under change of measure via conditions on...
Let $(\mathcal{X}, \mathscr{X})$ be a measure space and let $X:\Omega \to \mathcal{X}, X^\prime:\Omega \to \mathcal{X}$ be two random variables with probability measures $\mathbb{P}_X$ and...
View Article$\mathbb{E}(X^2) = 1$ implies the characteristic function is equal to $1$....
We let $X$ be a real random variable. And the characteristic function for $X$ is the function $\varphi: \mathbb{R} \rightarrow \mathbb{C}$ defined by $\varphi(u) = E(e^{iuX})$ for all $u \in...
View ArticleIt is the case that $E[\exp(X)] = \exp(E[X])$? [closed]
Is it the case that $E[\exp(X)] = \exp(E[X])$, where $X$ is a random variable? I know this is too simple, but I must be googling the wrong things.
View ArticleDoes convergence in probability transfer between absolutely continuous...
Given I have two random variables $X$ and $X^\prime$ defined on the same measurable space that are both absolutely continuous with respect to each other. What extra conditions do I need such...
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 ArticleCalculating the average value of a multi-variable function for one of the...
I have a multi-variable function of the form:$$F = F (r, \theta, \epsilon, \omega)$$where:r is the radial distance.$ \theta $ is the azimuthal angle, ranging from 0 to $\frac{\pi}{2}$.$\epsilon$ and...
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 ArticleExpected average of differences between two points in a uniform distribution
Let's say I have a discrete uniform distribution where a variable x can take any value between 1 to 100 (inclusive).I spin up two values of x: x1 and x2 and they take the values 23 and 53...
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