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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Second moment of weighted average of random variables

I stumbled upon problem 254 from the SOA Exam P list in https://www.soa.org/globalassets/assets/Files/Edu/edu-exam-p-sample-quest.pdf for which I am puzzled by the solution described in https://www....
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analytical asymptotic approximation of the expected maximum, mean, and minimum distance of nearest neighbours in unit ball

Say I uniformly at random distribute $x = n^3$ (independent identically distributed) points in a ball of radius $r=1$ in $\mathbb{R}^3$. What can be said about the expected maximum, minimum, and mean ...
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how can predictive distributions be considered as expectations?

I guess that the prior and posterior predictive distributions can be considered expectation of $p(y|\theta )$ (in case of prior predictive distribution) and $p(\widetilde{y}|\theta )$ (in case of ...
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Expectation of binomial random variable

Having trouble understanding something I read in a paper recently. Say we have $X \sim \operatorname{Binomial}(N,p).$ The paper states: $$E[X \mid N,p] = Np$$ (so far so good) and $$E[X] = \mu p$$ ...
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How to calculate the expectancy of the ratio of non-independent random variables?

How can I calculate this expectancy: $$E \left [ \frac{\sum_{t=1}^T{Z_tX_t}}{\sum_{t=1}^T{Z_t^2}} \right ]$$ where $Z_t \sim N(0,1)$ and $X_t \sim N(0,1)$ are independent? Any tricks? Is it ...
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Example in which $E[E[Y|T,X]] \neq E[Y|T, E[X]]$

Context: this question is a follow-up of this other question in which I'm trying to understand why we should use methods for causal inference instead of just training machine learning regressors, ...
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properties of a expectation for a non-negative random variable

Say I have a non-negative discrete random variable $X$ (values of $X$ can be mapped to integers $(0, 2^n -1)$ for $n \in \mathbb{Z}$) and an associated distribution $P(X)$. Given a non-negative scalar ...
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Scaling the conditioned random variable does not change conditional distribution, why?

Given two random variables $X$ and $Y$, I know intuitively that $$\mathbb{E}[X\,|\,Y]=\mathbb{E}[X\,|\,cY],$$ where $c$ is some non-random constant. My intuition tells me that scaling the ...
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Expectation of Mahalanobis Distance and its logarithm

Suppose: $$X \sim \mathcal{N}(X, \mu, \Sigma_x) \text{ st. } \Sigma_x \sim \mathcal{IW}(\Sigma_x; \Psi, v)$$ Where $\mathcal{IW}$ is the Inverse-Wishart distribution. This ...
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Describing guaranteed profit situations which are stronger than just 'superfair wager'

Context. I am tutoring a final year secondary school student in mathematics. To illustrate the principles of card-counting in a situation of sampling without replacement, I've decided to show her a ...
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Let $X$ be a positive, lognormal random variable with known mean $\mu_X$ and variance $\sigma_X^2$. Since $X$ is a lognormal random variable, I know its pdf and moment-generating function (mgf). pdf: $... • 143 0 votes 0 answers 20 views Question on the proof step in the theorem 1 of the Gap statistic paper From the Gap statistic paper, during the proof for the theorem 1, we can see the below equality (p. 422),$\begin{aligned} \operatorname{var}(X) & =\frac{1}{2} \int_{-\infty}^{\infty} \int_{-\...
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I'm having doubts on the conditions of an equality. Considering $f(X,w): \mathbb{R}^n \xrightarrow{} \mathbb{R}^n$ a function of n-variate random variable $X$ with an unknown distribution $p(x)$ and \$...