Mathematics Stack Exchange
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Questions tagged [expected-value]

Questions about the expected value of a random variable.

The average value of a randomly chosen quantity is its expectation or expected value. For example, the expected value of the number you get when you roll a fair 6-sided dice is 3.5.

In general, if $X$ is a random variable defined on a probability $(\Omega, \Sigma, P)$, then the expected value of $X$, denoted by $E[X], \langle X \rangle,$ or $\bar{X}$ is defined as the Lebegue integral

$$E[X]= \int_{\Omega} X(\omega) dP(\omega)$$

The expected value is often the first and most important thing you want to know about a random variable. For example, in a betting game, the best strategy is often the one that maximizes the expected value of the amount you win.

This tag is for questions about:

  • Computing the expected value in a specific situation.
  • Understanding the properties of expected values, such as Markov's inequality or linearity of expectation.
  • Proving theorems about the expected value of abstract random variables.
  • Understanding what the expected value means and what it tells you about a random variable.
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Why does the squared design matrix 1/N·X'X of the OLS converge to E(xx')?

We were told, that under the assumption that $y=\mathbf{x}'\beta_\circ+u$ in the representation of the ordinary least squares estimator (with $N$ being the sample number) $$\hat{\beta}=\beta_\circ + (1/N \cdot\mathbf{X'X})^{-1} (1/N…
Jaleks
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Deriving the expectation of a geometric with inverse CDF

Say I have a number $a$ and I want to find a number $b>a$ where $b \sim U[0,1]$. The value of b is a geometric random variable with well known expectation $\frac{1}{1-a}$ However, consider the number of draws to be a random variable $X$. We want…
Jason
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Expectation of a dt term

I am working with Hamilton-Jacobi-Bellman Equations and the following result appeared. Suppose $V_t= \frac{\partial V}{\partial t}$ and that $ E_{t}[Y]=E\left[Y \mid F_{t}\right] $ , where $F_t$ represents the information at time t I do not…
WHN
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Expectations of Probability Generating Function

I know that $$E(X) = P'_X(1)$$ then what would $$E(X(X-1))$$ formula would be?
Xenotion
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Does the equality $\sigma(cX)=\sigma(X)$ holds where $c\neq 0$ and $x$ is r.v?

I think the answer is yes because if $c>0$ then $\sigma(cX)=\sigma\{(cX\leq x)|x\in \mathbb{R}\}=\sigma\{(X\leq \frac{x}{c})|\frac{x}{c}\in\mathbb{R}\}=\sigma(X)$ and in the same manner if $c<0$ then $\sigma(cX)=\sigma\{(cX\leq x)|x\in…
Iesus Dave Sanz
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Minimizing the error of an approxmating function

Consider the problem of approximating a continuous analog signal X with a discrete digital signal Y. To do this, we may divide the range of X into discrete intervals, [a0,a1), [a1,a2), etc. Then, if X takes on a value in the interval [a0,a1), we…
Yihang
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Expected size of a set A

Let $I=\{1,2,\ldots, 1000\}$. Let $A$ be an empty set. I take a random integer from $I$ with replacement and put in $A$. I do this process 1000 times. What is the expected size of $A$? This is my approach. Let $X_i$ be binary random variable which…
user12290
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Does ${E}[\max\{X, Y\}] \geq {E}[\max\{X, Z\}]$ imply $\max\{{E}[X], {E}[Y]\} \geq \max\{{E}[X], {E}[Z]\}$?

Does $$\operatorname{E}[\max\{X, Y\}] \geq \operatorname{E}[\max\{X, Z\}]$$ imply $$\max\{\operatorname{E}[X], \operatorname{E}[Y]\} \geq \max\{\operatorname{E}[X], \operatorname{E}[Z]\}$$ (under the assumption that all the expectations are well…
Max
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Expected value of sample standard deviation

I am trying to prove that for i.i.d $X_1,X_2,...$ with expected value $\mu$ and variance $\sigma^2$, the sample standard variance $S^2=\sum \frac{(X_i-\overline{X})^2}{n-1}$ as expected value $\sigma^2$ where $\overline{X}=\frac{X_1+...+X_n}{n}$. I…
nagnag
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Division of random variable and condition inside an expected value

I have $\epsilon_0 \sim N(0, \sigma_0)$ and $\epsilon_1 \sim N(0, \sigma_1)$ are independent and $\epsilon_1 - \epsilon_0 = \nu$ How do you go from $E\left(\epsilon_0 | \nu \right) = \frac{\sigma_{0\nu}}{\sigma_\nu^2}\nu$ to…
Rami
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Expectation of a fraction form

Given $X \sim \exp(\theta)$ and $\bar{X}=\sum_{i=1}^nx_{i}$, how do I find $$\Bbb{E}\left[\frac{1}{\bar{X}-2}\right]?$$ Am I allowed to do this? $$\Bbb{E}\left[\frac{1}{\bar{X}-2}\right]=\frac{1}{\Bbb{E}[\bar{X}]-2}.$$
CJC .10
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What does this expectation symbol mean? Is it just the usual expectation?

I am feeling a bit silly for asking this question, but I am having trouble understanding this notation for (iterated?) expectation wrt multiple random variables used in the book "Fast algorithms via Approximation theory" by Vishnoi and Sachdeva…
smilingbuddha
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Expected Value $AB*CD$

The digits 1, 2, 3, and 4 are randomly arranged to form two two-digit numbers, AB and CD. For example, we could have $AB = 42$ and $CD = 13$. What is the expected value of $AB * CD$? On Brilliant, there is an explanation of this problem, but I…
Mining
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Finding expected value (AP + GP)

I am find the expected value for something and I have simplified it to the below, but I am not sure how to further solve it. Can soembody help and see if I am on the right track? My working: https://i.stack.imgur.com/Ftwo1.png I converted the 1 + 2…
Iva
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Expected number of Normals random number larger than the previous numbers

The problem assumes we randomly sample from a standard normal for 10 times. And ask the expected number of cases that the newly sampled number is the highest. It seems the relevant question has been discussed for the uniform, and the result has…
WWSS
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