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Questions tagged [expected-value]

The expected value of a random variable is a weighted average of all possible values a random variable can take on, with the weights equal to the probability of taking on that value.

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How do you derive a risk function from a loss function given a normal distribution?

Let X1....X25 be a random sample from a normal distribution N(p,1) such that the domain of p stretches from negative infinity to positive infinity. Let y = X-bar (I.e. the sample mean) Let loss ...
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Expectation of Multivariate Function

Let's say $p$ is a function of two, possibly correlated, other random variables $a$ and $w$ per the function $f()$: $p = f(w , a)$ What assumptions are necessary to express $E[w|p]$ as a function of ...
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Expected value as an orthogonal projection

I'm reading a paper in which the expected value of a random variable, $\mathbb{E}[X]$, is characterized as an orthogonal projection. This is on page 10. I've seen the geometric interpretation of ...
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Bayesian Inference and MSE. Need help to understand solution

I have problems with understanding solution of Problem 4.c (MSE) here. I couldn't get exact number. My solution is following (numbers for $X_M$ in table above): I start with calculating $E(X-X_M)^2 ...
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probabilistic model from expectation of another probabilistic model

In Goodfellow's deep learning book Chapter 13 first paragraph (https://www.deeplearningbook.org/contents/linear_factors.html) Many of the research frontiers in deep learning involve building a ...
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Expectation of a random walk that can't go below zero

Suppose we have a random walk $S_n$ that is constrained to be positive or zero, that is: $$S_0 > 0$$ $$S_{i+1} = \max(S_i+x_i,\space 0)$$ $$x_i \sim N[\mu,\sigma^2]$$ Can we analytically ...
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Expected Return to the Origin – Interpreting an expectation formula

I am trying to get a head start on the next semester at uni. The following question is based on the statistical problem and solution outlined on pages 3 to 5 of this book. The problem is based on ...
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23 views

Expectation of product - formula check

if $x_i$ are IID continuous random variables, with $E[x_i]=μ$, is the following correct? $E\left[\prod_{i=1}^n(1+⍺\space x_i)^i\right] = \prod_{i=1}^nE\left[(1+⍺\space x_i)^i\right] = \prod_{i=1}^...
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Confusion about step in deriving Bellman equation from value function

I am reading Reinforcement Learning, An Introduction by Sutton, Barto and I came across the derivation $$ \begin{align} v_{\pi}(s) &= \mathbb{E}_{\pi}\left[ G_{t} | S_{t} = s \right] \\ &= \...
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1answer
20 views

Analytical solution to the covariance between a continuous and a categorical variable

Let $X$ be a continuous variable with mean $\mu$ and $Y$ be a categorical variable with event probability vector $\mathbf{p}$. I am trying to calculate $Cov(X, Y)$. I have the solution if $\mathbf{p} ...
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For which distributions of X, the expected value E[X] equal to arithmetic mean? [closed]

For which distributions of X, the expected value of X (i.e., E[X]) is equal to the arithmetic mean of X?
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In a test that has multiple choice questions with four options each, when should a test-taker not pick options randomly?

Let's say that a test-taker takes a test with multiple choice questions with four options. Picking the right answer 1 mark and there is a penalty of 0.25 marks (say) for picking the wrong answer. That ...
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1answer
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Difference between averaging and ignoring the partial dependencies?

This question sparks from model interpretation/visualization. To graph the dependency of a function with >2 arguments, one often needs to ignore or average out some arguments. Problem set Hastie, ...
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49 views

Cramér-Rao Lower Bound & Fisher information - error in textbook?

I'm currently reading the textbook "Statistics for Mathematician" from Victor Panaretos. On page 65, the author presents the following equation for the Cramér-Rao Lower Bound (Note: I set the ...
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1answer
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Expectation of reciprocal of $(1-r^{2})$

I tried finding expectation from the density function but then realised that I was solving with the density function of $r$ and not it's square. I don't know the density function of $r^{2}$. I am ...
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44 views

Calculating Conditional Expectations

Say $X$ is continuous and distributed according to CDF $G$. How would we calculate $\text{E}[X|X<a]$?
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Contradiction between memorylessness of random walk to straight forward calculation

A drunk person is in place $x=0$ in time $t=0$. Every second he moves forward one meter in a probability of $\frac{1}{2}$ or stays in his place in a probability of $\frac{1}{2}$ (his decisions are ...
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intuitive explanation for expected value of the square of a uniform variable

I'm confused about something that should be simple. Suppose I have a random uniform variable $X$ on $[0,1]$. It's fairly clear that the expected value of $X$ is 1/2. By integrating $x^2$ on $[0,1]$, I ...
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1answer
36 views

Expectation of norm ratio

Suppose $x = (x_0, x_1, \ldots, x_{n-1}) \in \mathbb{R}^{n}$ with $x_{i} \sim \mathrm{Unif}(0, b)$. How can I calculate or estimate $E\left[\left\|x \right\|_{1}/\left\|x \right\|_{2}\right]$? I ...
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Expectation of a discrete random variable that is case-defined from other discrete random variables

Background The question arises from the following real-life situation: I buy a newspaper at 3 dollars and sell it at 6 dollars. I know the demand for news paper is a binomial random variable with $n=...
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Expected value for classifier evaluation (confusion matrix, expected rates and cost/benefit matrix)

I am currently doing a machine learning project based on historical data of a game. I am trying to predict which team will win, i.e. team one or team two, based on in-game happenings. The target value ...
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Expected value of statistic based on sample correlations

Let there be $i = 1,...,m$ random variables, for simplicity assume that each of these random variables follows a normal distribution: $x_{i} \sim N(\mu_{i}, \sigma_{i})$. Let $\hat{\rho_{i,j}}$ be the ...
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Covariance of Two Quadratic Forms

We're looking for the $\operatorname{Cov}\left[x^T A x, ~x^T B x\right]$ where $x$ is random variable and mean-centered, but not independent and $A$ and $B$ are symmetric matrices. The fundamental ...
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Expected value of weighted random variable

I have a statistic I'm investigating that depends on estimates from a correlation matrix. The statistic is: $$ T = \sum_{i=1}^{m} d_{i} w_{i} $$ where $d_{i}\sim Bern(\pi)$ and is independent of $w_{i}...
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Finding the expected value of sum of squares

I am still confused about how to get the expected value of sum of squares from a statistical model. For example, for the model: $y=\tau_i+\beta_{j(i)}+ (\tau\beta)_{ij}+\gamma_{k(j)}+(\tau\gamma)_{...
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Sum of the Expected Value of a Quotient. Is putting the Sample Size equal to 1 allowed?

Let $Y_1,...,Y_n$ be independent and identically distributed (i.i.d.) (standard/regular) inverse Gaussian random variables with parameters $\mu>0$ and $\lambda>0$. It is given that $E(Y_i)=\mu$. ...
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1answer
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Soft thresholding (Donoho and Johnstone)

Donoho and Johnstone (1994) poses the following equality: $$ E((\eta_t(X) - \mu)^2) = 1 - 2\Pr(|X|\lt t) + E(\min(X^2,t^2)) $$ where $\eta_t(X) = \operatorname{sign}(x)\max(|X|-t,0)$ and $X \sim N(\...
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25 views

In multivariate error function: why $\frac{1}{2n}$ in $E(w)=\frac{1}{2n} \|Xw -t \|_2^2$?

In multivariate error function: why $\frac{1}{2n}$ in $E(w)=\frac{1}{2n} \|Xw -t \|_2^2$? What does it do? What is it related to?
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72 views

Expected value notation

I come across the notions $\mathrm{E}[wage]$ and $\mathrm{E}[wage_i]$. I would like to clarify if they have different meanings. Take the first expression. $\mathrm{E}[wage]$ could represent the ...
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What is the expectation of $\log(1+aX)$ where $X$ has an $F$ distribution?

The random variable $Y$ is defined by $Y=\log(1+aX)$ where $X$ has an $F(m,m)$ distribution and $a$ is a non-negative constant. What is the expectation of $Y$?
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Calculating statistics on transformations of many random variables

I have a model that uses $n$ independent random variables $X_1,..., X_n$. I know the density function of each random variable. I would like to calculate statistics such as $E(\sqrt{X_1+...+X_n})$ or $...
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Tight upper bound on the expectation of a concave function

N is a random variable whose sample space is [0,$\infty$). I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The ...
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How to derive $\operatorname{var}[(X_i−\mu)^2]=2\sigma^4$ where $X$ is distributed normally

I have $X_1,...,X_n$, i.i.d. $N(\mu,\sigma^2)$ and I would like to calculate $\text{var}[(X_i−\mu)^2]$. I know that the solution is $2\sigma^4$. However, I can't derive it. Any suggestions?
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Prove convergence in distribution, probability, or quadratic mean for a sequence of binary variables that depend on another binary variable

Suppose that $X$ has the support set $\{1, -1\}$, and $P(X = 1) = P(X = -1) = 0.5$. Suppose that $X_n$ has the support set $\{X, e^n\}$, and $P(X_n = X) = 1 - \frac{1}{n}$ $P(X_n = e^n) = \frac{1}...
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Expectation of Inverse Logit of Normal Random Variable

I have a random variable $Y = \frac{e^{X}}{1 + e^{X}}$ and I know $X \sim N(\mu, \sigma^2)$. Is there a way to compute $\mathbb{E}(Y)$? I have tried to work out the integral, but haven't made much ...
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1answer
31 views

How to minimise $\mathbb{E}_X[\log(\text{sigmoid}(x + y))]$?

I want to know whether I can obtain the closed form of the minimising value: $$\text{argmin}_y \mathbb{E}_{x \sim p(x)}[\log(\text{sigmoid}(x + y))],$$ where the sigmoid function is $\text{sigmoid}(...
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1answer
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Finding joint probability distributions from marginal distributions

Question: I was solving test papers where I found this one. My doubt: I know to work with conditional probabilities and Jaccobian Transformation and part A and B can be done applying the above..But ...
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Relation between Expectations of sum of Bernoulli variables

Let $X_1, ... X_n$ and $X_1^+, ... X_n^+$ be two finite sequences of non-independent, non identically distributed Bernoulli variables such that $E[X_i^+] \geq E[X_i]$. If we define $S = \sum_{i =1}^n ...
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Expected Value: Decimal or Whole values?

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 ...
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1answer
99 views

Expectation of E(g(X + c))

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 ...
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Expected value of quotient of Poisson distributions

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 ...
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1answer
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How to calculate expected risk from fitted Cox PH model in R?

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,...
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1answer
55 views

Show $E[ (Y - E(Y|X)) (E(Y|X) - h(X))] = 0$

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 ...
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60 views

Multiple interval ratio of E[X / (X + Y)]

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-...
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Batch Normalization expectation operator

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. ...
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Why is expected value of random variable equal to mean

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 ...
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Variance of continuous Random Variable negative?

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 ...
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Expectation of inverse non central chi-squared

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^{...
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1answer
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Comparing suprema of inner products of Gaussian variables

I'm given two i.i.d. standard normal vectors $x, y \sim \mathcal{N}(0, I_n)$, and vectors $a \in \mathbb{S}^{n-1}$, the unit sphere in $n$ dimensions. Additionally, given a set $S \subseteq [n]$, I ...
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Expectation of the minimum of dependent random variables

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,\ ...