Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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.

Overview

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. For a discrete random variable, $X$, the expected value is

$$ E(X) = \sum_{x} x P(X=x) $$

for a continuous variable with probability density function $p(x)$,

$$ E(X) = \int_{-\infty}^{\infty} x p(x) dx $$

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Sample two numbers from 1 to 10; maximize the expected product

Assume you sample two numbers, randomly drawn from 1 to 10; you could choose two strategies: 1) pick with replacement and 2) pick without replacement. Which strategy would you prefer to maximize the expected product? I encountered this problem while…
user334639
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Expectation of reciprocal of a variable

I am confused in applying expectation in denominator. $E(1/X)=\,?$ can it be $1/E(X)\,$?
Shan
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Relation between "expectation of ratio" and "ratio of expectation"

For two random variables $A$ and $B$. Often times, I say people write the following, \begin{equation} E(\frac{A}{B})=\frac{E(A)}{E(B)}\{1- \frac{Cov(A,B)}{E(A)E(B)}+\frac{Var(B)}{[E(B)]^2} \}. \end{equation} Can someone give me a clue how to derive…
Jie Wei
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Is probability equal about the mean?

Suppose that $X$ is a random variable with PDF $f(x)$ with support $(-\infty,\infty)$. Suppose that the expectation of $X$ is $\mathbb{E}(X)=\lambda$. Is it always true that $$\int_{-\infty}^{\lambda} f(x)dx =…
EssentialAnonymity
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How do you calculate the expected value of $e^{-X}$?

If $X$ is a random variable, I would like to be able to calculate something like $$E(e^{-X})$$ How can I do this? Thank you so much.
anxoestevez
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Questions about antithetic variate method

Suppose we are to estimate a expectation problem $E(f(X))$, where $X$ is a random variable with known distribution, by simulation and Large Law of numbers estimator. Antithetic method is a way to reduce variance of estimator in such cases. If $X$ is…
Tim
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Number of expected pairs in a random shuffle

Suppose there are 34 poeople standing in a row in random order, among them 18 are male and 16 are female. If two people adjacent to each other belong to different genders, we consider them to be a couple, how many couples are we expected to see on…
Xiaowen Li
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expectation of an exponential function

What is the expectation of an exponential function: $$\mathbb{E}[\exp(A x)] = \exp((1/2) A^2)\,?$$ I am struggling to find references that shows this, can anyone help me please? I am assuming Gaussian distribution. A is a constant and x is a random…
Damian
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In which cases we can approximate expected value of a function by assuming the function and the expectation commute?

I was reading a computer vision paper and the authors approximated $\left_{Q}$ with $f( \left< X \right>_Q)$ where $f(\cdot)$ is nonlinear. Are there any rules of thumb for this kind of approximation? In which cases we can use such an…
nimcap
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Expected value of dot product between a random unit vector in $\mathbb{R}^n$ and another given unit vector

I am wondering what is the $\mathbb{E}[(x\cdot v)^2]$ where $x$ is a random unit vector in $\mathbb{R}^n$ and $v$ is a given unit vector in $\mathbb{R}^n$. By $(x\cdot v)$ I mean the dot product between $x$ and $v$. I read somewhere that…
Learner
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Calculating expected return

Assume I have a trading system that I'm evaluating over a three-year period. The returns are 25%, -40% and 25%. Empirically, I can see that this system loses because at the end of three years, I have less than when I started. Wikipedia defines…
Milktrader
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Expectation of $\left(X-M\right)^T\left(X-M\right)\left(X-M\right)^T\left(X-M\right)$

If $X=[x_1,x_2,...,x_n]^T$ is an $n$-dimensional random variable and we have $E\left\{X\right\} = M = \left[m_1,m_2,...,m_n\right]^T$ $Cov\left\{X\right\} = \Sigma = diag\left(\lambda_1,\lambda_2,...,\lambda_n\right)$ how can I express the…
Isaac
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When is $\mathbb{E}\left[\frac{1}{\sum X_{i}}\right] = \frac{1}{\mathbb{E}\left[\sum X_{i}\right]}$?

Are there specific conditions under which the following is true? E.g., certain distributions, positive RV? $$\mathbb{E}\left[\dfrac{1}{\sum X_{i}}\right] = \dfrac{1}{\mathbb{E}\left[\sum X_{i}\right]}$$
User0
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Expected value of $h(X)$. When can the order of $E$ and $h$ be inverted?

This doubt arose when dealing with the typical exercise of calculating the expected grade of a multiple-choice exam answered at random (where each right answer is given $p_1$ points and each wrong answer gets $-p_2$ points). Let $X \sim…
Vicent
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Voting Statistics Problem

There is a statistics bonus problem I had on an exam, and I got it wrong. It is as follows: There is a county in which 100, 000 people vote in an election. There are only two candidates on the ballot: A and B. In this county, 70, 000 people go to…
John Yils
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