expected-value's questions - English 1answer

1.127 expected-value questions.

This is from Time Series Analysis from WS Wei If, where Approach - We have to find Var(Z) So, since and I have simulated this scenario on computer with and the answer seems to be ~ 0.5 ...

Say you are given a sample size consisting of a number of whole objects, such as cats (n=20). You are asked what the expected value of some factor of these cats is (ex. how many are expected to be ...

Let $X$ be a random variable, and let $f$ be a concave function. Are there any known lower bounds for (or methods of lower bounding) $\mathbb{E}[f(X)]$? Jensen's inequality only gives an upper ...

In the barto and sutton book, the authors have provided an Example on page 106, Ex 5.5 where they prove that the variance is infinite for an ordinary importance sampling method. In this derivation, ...

I am curious what the derivation for the expectation of the maximum of two jointly normal random variables $X$ and $Y$ with correlation coefficient $\rho$. I could start with the following but the ...

I was doing some work in scipy and a conversation came up w/a member of the core scipy group whether a non-negative discrete random variable can have a undefined moment. I think he is correct but do ...

Can there exist two random variables such that - 1) Their means are not equal. 2) Their k-th order raw moments for all k>1 are equal.

Let $X$ and $Y$ be independent random variables such that $X \sim \text{Poisson}(\lambda \cdot c)$ and $Y \sim \text{Poisson}(\lambda \cdot (1-c))$, where $c$ is a real number in $[0, 1]$. Is there ...

Take an expectation of the form $E(f(X))$ for some univariate random variable $X$ and an entire function $f(\cdot)$ (i.e., the interval of convergence is the whole real line) I have a moment ...

The random variable $X$ assumes values 1; 2; 3 with probabilities: $P(X = 1) = \theta^2$ $P(X = 2) = 2\theta (1-\theta ),\quad 0 < \theta < 1$ $P(X = 3) = (1 -\theta)^2$: If in a random sample ...

I am little confused regarding finding expectation of vector multiplied matrix. X is vector having mean $m_x \in \mathbb{R}^{N\times 1}$ and variance vector $s_x \in \mathbb{R}^{N\times N}$. $\phi$ is ...

I have a sequence of interchanging on- and off-intervals, each pair identified by index $i$. The duration of the on-interval $i$ is represented by random variable $X_i$, and the duration of the off-...

I have a question. A and B are normal distibutions. how to calculate $\operatorname{E}_{A,B}[A^2]$ does it scene that I drop the B in the statement or do I have to consider the B somehow ?

Say I have an unbiased coin and if I roll heads I get 40 pennies reward and If I roll tails I get 80 pennies. I believe the following is correct for the expected reward from one toss: ...

We probably have played the game "Throwing Balls into the Basket". It is a simple game. We have to throw a ball into a basket from a certain distance. One day we were playing the game. But it was ...

$\newcommand{\E}{\mathbb{E}}$Let a finite collection of exchangeable random variables $X_1,...X_n$ (some authors would call this collection "interchangeable" since it is finite, reserving "...

Trying to understand the result of an equation at pages 129-130 in the book Hamilton - Statistics in physical science but I seem to be missing something. $$ E(V^TPV) = E[(F-F^0)^TP(F-F^0)] - E[(\...

Show that $E[ (Y - E(Y|X)) (E(Y|X) - h(X))] = 0,$ where $X, Y$ are random variables with constant means and $h(x)$ is an arbitrary function. So far, I have expanded out the expectation and used ...

I am asked to find the expected value of a vector of two random variables when the joint density is given. Is the recipe for solving this problem: Find the marginal distributions Find the expected ...

I'd like to calculate expected risk (cumulative incidences), which are derived from fitted Cox PH model using R packages. I have the fitted Cox PH model like as follows: [Variables] Dataset: 10,...

From 'Modern Mathematical Statistics with Applications' (Devore and Beck) pg 377 Let $X_1, X_2 \ldots$ be a random sample from the disbribution $f(x,\theta) = \theta x^{\theta-1}$ for $x\in {]0,1[}$...

I am looking to measure agreement between participants in choosing which member of a group is most like certain attributes. I want to calculate the expected agreement if they were to choose by chance. ...

If X is a continuous random variable, under what conditions is the following condition true E[|x|] = E[x] ?

I have independent variables $X_i\in[0;1]$ and suppose they are uniformly distributed. If you want to minimize the total absolute deviation to a fixed number, how much can you gain from using the ...

A derivation in a paper (theoretical ecology--there are often mathematical errors there) I am reading essentially uses the following line: $\frac{1}{n}\sum_{i=1}^{n}X_{i}=E\left[X_{i}\right]$. This ...

I am working through an expectation and have something that I want to be true, and appears to be true in simulation, but I am having a hard time writing a proof that the idea can be derived properly. ...

For the two sample t-test with unequal variances: $T=\frac{\bar{x}-\bar{y}}{s_x^2/n_1+s_y^2/n_2}$, we usually use welch approximation, thus $T\sim t(f)$, where $f=\frac{(\frac{\sigma_x^2}{n_1}+\frac{\...

I understand how to get E(X), but how can I derive E(2^X)?

While learning about Random variables I came across the mean of random variable X. The definition says that the expected value of random variable E(X) = Mean of Random variable X I am not able to ...

On page 2 the paper Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift they pull a couple of fast moves with the expectation operator and I'm not sure why. ...

How do we compute the expectation of the minimum of dependent random variables? In other words, what is the value of $\mathbb{E}[Y]$ in the following case: $$ \mathbb{E}[Y]= \mathbb{E}\big[\min(X_1,\ ...

I am having trouble finding the Variance for this question. The proportion of salt X left in the salt shakers at the end of the day at a crowded restaurant has a probability density function given ...

In the context of likelihood-based inference, I've seen some notation concerning the parameter(s) of interest which I've found a little confusing. For example, notation such as $p_{\theta}(x)$ and ${...

If $X\sim \mathcal{N}_p(\theta,\sigma^2)$ then $X^{'}X/\sigma^2\sim$ non-central $\chi^2_p$ with non-centrality parameter $\theta^{'}\theta/\sigma^2$. How to find out the expectation of $\sigma^2/X^{...

I have a fairly basic question that I'm looking for a reference for. First, a couple definitions. Let's say $X_1,\ldots,X_n$ are IID samples from a distribution $F$ over $[0,1]$. For any $k\in\{1,\...

I am working through past examination questions from the Royal Statistical Society and came across this one from 2009 in Module 5 (Question 2(i) and Solution): The random variable $X$ has a $\chi^{...

Does anyone know of a good resource listing known tricks (with examples?) for calculating closed form expressions from messy expectations? (e.g., moment generating function, law of iterated ...

I saw the answer on this post and got confused about a couple things in its explanation. Mainly, I am unsure of How the poster immediately knows the process $X_t = c+\phi_1 Y_{t-1} + \epsilon_t$ is ...

Are there situations when it is allowed to omit calculating expected agreement but use only observed agreement as reliable measure? I have multi-label classification (in particular annotation of ...

Let $X_1,\dots,X_n \sim U(0,1)$ be independent and identicallly distributed standard uniform random variables. $$\text{Let }\quad Y_n=\sum_i^nX_i^2 \quad \quad \text{I seek: } \quad \mathbb{E}\big[\...

First a note: I am not a statistician. I studied maths at university (but opted out of every single stats class), and now find myself in a job where I'm doing stats. My question is a little bit ...

I am struggling with the following exercise in the context of modeling information structure via filtration to evaluate contingent claims. I hope that someone can explain me how to derive the solution:...

If we flip a fair coin until we get heads, what is the variance of the number of flips to do this? My attempt is: $$E(flips):=Y=1\times P(H)+(1+Y)\times P(T)$$ $$\Rightarrow Y=\frac{1}{2}+\frac{1+Y}{...

I am implementing an Elo ranking system for fun and came across an example on Wikipedia that I can't follow: Suppose Player A has a rating of 1613, and plays in a five-round tournament. He or she ...

I would like to know what the formal definition of the following expression is $$ V_\pi(s) = \mathbb{E}_{\pi}(G_{t+1} | S_t =s) $$ What does it mean to have the policy in the subscript? How would I ...

How can I find an expectation of random variable $\xi \sim \mathcal{N}(a,1)$, where $a$ is random variable: $a \sim \mathcal{N}(0,\sigma^2),\quad \sigma = \text{constant}$ ?

Define $$\psi(x)=\begin{cases} 1-p & x < 0 \\ 0 & x=0 \\ -p & x> 0 \end{cases}$$. I have to show that if $$E\psi(x-\theta)= 0 $$ then $$P(X< \theta) \leq p \leq P(X \leq \theta)$$...

I am trying to read the Elements of Statistical Learning Tibshirani, Hastie and Friedman, however I have a problem with understanding the expected (squared) prediction error ($EPE$) formula that they ...

Say, there is a $n$-dimension multivariate Gaussian, $g(x) = N(x:\mu, \Sigma)$ where $\mu$ is $n$-dim mean vector, and $\Sigma$ is $n \times n$-dim covariance matrix. I would like to calculate "...

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