# 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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### Is There a Standard Metric for Evaluating Treatment Impact Considering Action Cost in Uplift Models?

I'm currently exploring Uplift modeling, specifically the use of the Conditional Average Treatment Effect (CATE) metric: $$\tau(t', t, x) := \mathbb{E}[Y | X=x, T=t'] - \mathbb{E}[Y | X=x, T=t]$$ ...
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### MCMC Intuition for Expected Value Estimate

I believe I have a fundamental misunderstanding of MCMC. We seek to estimate the expected value of a target distribution using its integral form. We consider a markov chain of samples from a simpler ...
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### Estimating expected value with respect to posterior

I have a neural network and I need to calculate the following: $$\mathbb{E}_{P(\theta|D)}[f(\theta)]=\frac{\sum_\theta P(D|\theta)P(\theta)f(\theta)}{\sum_\theta P(D|\theta)P(\theta)}$$ Where $f$, ...
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### What is the expected length of an interval on an arc of a circle that can be constructed using exponential variates?

I had asked this question on Math stackexchange once before and now again but this does not seem to be drawing too much attention. Since this is a question that can be safely classified as non-measure ...
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### Calibration Expectation Decompostion

I am reading a "Calibrated Structured Prediction by Kuleshov and Liang" link. Calibration and sharpness. Given a forecaster $F : X → [0, 1]$, define $T(x) = \mathbb{E}[y| F(x)]$ to be the ...
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### Expected Value / Variance of Gamma to the Negative Integer

For random variable $Z$ from a $Gamma(p), p > 0$ distribution we know that the expected value of $E[Z^s]$ is simply the gamma function at $p+s$ divided by the gamma function at $p$, for $p+s$ > ...
1 vote
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### What is the fourth moment of a Euclidean Norm?

Let $X=\lVert M^\top p\rVert_2$, where $M$ is an $n\times n$ non-random matrix and $p\sim N(0,I_{n\times n})$ is an $n\times 1$ vector, and$\lVert \cdot\rVert_2$ is the Euclidean norm. Using some ...
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### Baffled by Rubin's Potential Outcomes RE: What Would Have Happened?

Background Seeking a clear and authoritative explanation of a key concept of Rubin's Potential Outcomes Framework that is causing this hapless OP enormous grief. While the necessity to distinguish ...
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### How to tell if an estimator is unbiased? How to find expected value of an estimator?

You come up with a great idea of an estimator for $\beta_1$ in the SLR model which satisfies SLR.1 to SLR.4:$$y_i=\beta_0+\beta_1x_i+u_i$$ Given a sample $\left\{(x_i,y_i),i=1,2,3,\dots,n\right\}$, ...
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### Expected number of elements in random bucket, where the probability of a bucket is proportional to its size

I distribute n balls into m buckets randomly and uniformly. Let's assume that n/m is greater than 1, and m is a large number. Next, I choose a bucket at random, but the probability of selecting a ...
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### Between steps for fisher information matrix element using Poisson regression?

I am currently working through some math related to my work, and trying to understand how the individual pieces of the following equations come together for the Fisher information matrix expression (...
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### Variation on St. Petersburg Paradox, with total loss at the end

I’m not too sure how to answer these variations I thought of and was hoping someone could enlighten me. A casino offers a game of chance for a single player in which a fair coin is tossed at each ...
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### Let $N(t)$ is a poisson process with rate $\lambda$, $T^* \sim \operatorname{Exp}(\lambda^*)$, find the expectation of $N(\min(t, T^*))$

Currently, my approach is to split $N(\min(t, T^*))$ like the following by the law of total expectation. \begin{align*} &E(N(\min(t,T^*))) = E(N(t \wedge T^*)) \\ = {}&E(N(t\wedge T^*) \...
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### Expected absolute deviation greater than standard Laplace

Could there exist a distribution, other than standard Laplace (probability density of the form $1/2e^{-|x|}$), on $\mathbb{R}$ such that $E[x]=0,E[|x|]=1$ and that \begin{equation*} E[|x-a|] \geq |a|+...