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 $p(x)$,

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

1985 questions
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Why is expectation the same as the arithmetic mean?

Today I came across a new topic called the Mathematical Expectation. The book I am following says, expectation is the arithmetic mean of random variable coming from any probability distribution. But, it defines expectation as the sum of product of…
pranphy
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Find expected value using CDF

I'm going to start out by saying this is a homework problem straight out of the book. I have spent a couple hours looking up how to find expected values, and have determined I understand nothing. Let $X$ have the CDF $F(x) = 1 - x^{-\alpha},…
styfle
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Taking the expectation of Taylor series (especially the remainder)

My question concerns trying to justify a widely-used method, namely taking the expected value of Taylor Series. Assume we have a random variable $X$ with positive mean $\mu$ and variance $\sigma^2$. Additionally, we have a function, say,…
agronskiy
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Deriving Bellman's Equation in Reinforcement Learning

I see the following equation in "In Reinforcement Learning. An Introduction", but don't quite follow the step I have highlighted in blue below. How exactly is this step derived?
Amelio Vazquez-Reina
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What is the difference between finite and infinite variance

What is the difference between finite and infinite variance ? My stats knowledge is rather basic; Wikipedia / Google wasn't much help here.
AfterWorkGuinness
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Brain-teaser: What is the expected length of an iid sequence that is monotonically increasing when drawn from a uniform [0,1] distribution?

This is an interview question for a quantitative analyst position, reported here. Suppose we are drawing from a uniform $[0,1]$ distribution and the draws are iid, what is the expected length of a monotonically increasing distribution? I.e., we…
Amazonian
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Why is the expected value named so?

I understand how we get 3.5 as the expected value for rolling a fair 6-sided die. But intuitively, I can expect each face with equal chance of 1/6. So shouldn't the expected value of rolling a die be either of the number between 1-6 with equal…
Nithish Inpursuit Ofhappiness
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Can somebody offer an example of a unimodal distribution which has a skewness of zero but which is not symmetrical?

In May 2010 Wikipedia user Mcorazao added a sentence to the skewness article that "A zero value indicates that the values are relatively evenly distributed on both sides of the mean, typically but not necessarily implying a symmetric distribution."…
Andy McKenzie
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MSE decomposition to Variance and Bias Squared

In showing that MSE can be decomposed into variance plus the square of Bias, the proof in Wikipedia has a step, highlighted in the picture. How does this work? How is the expectation pushed in to the product from the 3rd step to the 4th step? If the…
statBeginner
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Why not report the mean of a bootstrap distribution?

When one bootstraps a parameter to get the standard error we get a distribution of the parameter. Why don't we use the mean of that distribution as a result or estimate for the parameter we are trying to get? Shouldn't the distribution approximate…
Guillermo Perez
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Why should the frequency of heads in a coin toss converge to anything at all?

Suppose we have any kind of coin. Why should the relative frequency of getting a heads converge to any value at all? One answer is that this is simply what we empirically observe this to be the case, and I think this is a valid answer. However, my…
Maximal Ideal
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I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?

The title is the question. I am told that ratios and inverses of random variables often are problematic. What is meant is that expectation often do not exist. Is there a simple, general explication of that?
kjetil b halvorsen
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Why maximum likelihood and not expected likelihood?

Why is it so common to obtain maximum likelihood estimates of parameters, but you virtually never hear about expected likelihood parameter estimates (i.e., based on the expected value rather than the mode of a likelihood function)? Is this primarily…
Jake Westfall
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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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How do I analytically calculate variance of a recursive random variable?

Suppose I have a chest. When you open the chest, there is a 60% chance of getting a prize and a 40% chance of getting 2 more chests. Let $X$ be the number of prizes you get. What is its variance? Computing $E[X]$ is fairly straight forward: $E[X] =…
Brian
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