Expectation of Cosine Similarity of Random Normalised Vectors in High Dimensions
Given random zero mean, i.i.d, isotropic random vectors $X_1, X_2, X_3\in \mathbb{R}^n$ such that $\|X_i\|=1$ and $n$ is sufficiently high (e.g. n=512), let$$A = X_2 - X_1,\quad B=X_3 - X_2.$$What is...
View ArticleExpected Benefit
The question is the following:A five year term insurance policy pays 25000 if the insured dies in the first year. The benefit declines by 5000 per year for each of the next four years. In each of the...
View ArticleConnection between subgaussian and polynomial tail bounded variables.
I have $p$-dimensional independent vectors $\boldsymbol{X}_1,\ldots,\boldsymbol{X}_n.$ I have also condition: for some $\eta>0$:\begin{equation}\mathbb{E}\left(e^{tX_{ij}^2}\right)\leq K<\infty...
View ArticleConjecture: If $x_k$ are random in $(0,\pi/2)$ then expectation of...
Let $E(n)=\text{expectation of }\dfrac{\prod_{k=1}^n\tan x_k}{\sum_{k=1}^n\tan x_k}$ where $x_k$ are independent uniformly random real numbers in $\left(0,\frac{\pi}{2}\right)$.Is the following...
View ArticleExpectation of product of functions of random variables
If $X$ and $Y$ are independent continuous random variables (each withfinite expectation), the product of their expectation is the productof their individual expectation.This can proved by starting from...
View ArticleIf $X,Y \sim U(0,1)$, what is ${\Bbb E} (XY)$?
If $X,Y \sim U(0,1)$, what is ${\Bbb E} (XY)$? That is, what is the expected value of the product of two uniformly distributed independent random variables? It is probably not $\frac14$? What is the...
View ArticleCalculating the expected value and variance of sum of iid normally...
Let $X_1, X_2, ..., X_n$ be an iid sample from $N(\mu, \theta)$, where $\mu$ is unknown. I'm trying to find the expected value and variance of the random variable $Y = \frac{1}{n} \sum_{i=1}^n (X_i -...
View ArticleConditional probability distribution: interpreting expectation and $P(E_k)$...
Context: This is a problem I'm (still) coming up with for my math club meet, so there may be some ambiguities but please give me the benefit of the doubt;Suppose that an underdog table tennis player,...
View ArticleExpectation of multivariate function with fixed entry
I am given a random vector $\underline{X}$ of length n and a multivariate function\begin{align*}\Phi:\Omega&\to\mathbb{R}\\\underline{x}&\mapsto...
View ArticleEvaluating a binomial sum in expectation and understanding expectation
Original problem: Let $X$ denote the number of heads that appear in $n$ tosses of a fair coin. What is the mean/expected value and standard deviation of $X$? What is the probability that the number of...
View ArticleExpectation of $\mathbb{E}|y^*aa^*x|$ with Gaussian distribution?
For $\pmb{a}\sim\mathcal{N}(0,\pmb{I}_n)$ and $\|\pmb{x}\|=\|\pmb{y}\|=1$, one has $$\mathbb{E}|\pmb{y}^*\pmb{a}\pmb{a}^*\pmb{x}|=\frac 2 \pi(\sin\theta+(\frac \pi 2-\theta)\cos\theta)$$ by setting...
View ArticleExpected value of an inhomegenous Poisson process with random intensity?
Let $N(t)$ be a homogenous Poisson counting process with rate $\rho$.Let $N^*(t)$ be an inhomogeneous Poisson counting process with a time-dependent rate given by:\begin{equation}\rho^*(t) =...
View ArticleValue of this collatz constant? [closed]
Let $1<s$ be an integer.How many iteration paths $w(s)$ exist to get from $1$ to $s$ under the conditionsA) the values of the iterations do not go above $2s+1$B) The valid iterations are $2x$ and...
View ArticleConfusion over calculation of expected value of sample variance
I want to show that the sample variance $$S^2 := \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X_n})^2$$ is unbiased, i.e. $\mathbb{E}[S^2] = \sigma^2$.I know that the fastest way of showing this is by adding...
View ArticleValidity of expectation handling in "decoupling" Lemma proof
I am taking a course in Random Matrix Theory, and recently the lecturer has introduced us the following lemma, which is considered auxiliary before we move to concentration inequalities. Now, since the...
View Article$E[X_1+X_2+\cdots+X_n]=E[X_1]+E[X_2]+\cdots+E[X_n]$ Proof
I am trying to proof (from myself I have the case in my book for continuous random variable but want to find the proof for discrete random variables)...
View ArticleCalculate $\mathbb{E} \left[ \frac{X_k}{\sum_{i=1}^{N} X_i + || v \frac{A}{\|...
I want to calculate $\mathbb{E} \left[ \frac{X_k}{\sum_{i=1}^{N} X_i + || v \frac{A}{\| A \|_F}||^2} \right]$when$X_{i}s$ are independent but not identical exponential RVs$v$ is size $1 \times N$...
View ArticleComparing two random variables based on relation among their expectation...
Let X and Y be discrete random variables. If pr$(X \geq Y) \geq k$ I can easily infer that $\mathbb E (X)\geq k \mathbb E (Y)$. I am struggling to prove the converse but somehow it seems reasonable.
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 ArticleThe distribution of the ratio between Binomials [closed]
What is the distribution of $\frac{X}{X+Y}$ where 𝑋 and 𝑌 are Binomial(𝑛,(1-p)$\alpha$ ) and Binomial(𝑛,p$\beta$) respectively?
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