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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How to derive the expectation of $\log[a \theta_k + b]$ in Dirichlet distribution?
Given that $\boldsymbol{\theta} \sim \mathrm{Dir}(\boldsymbol{\alpha})$, then $E_{p(\boldsymbol{\theta} \mid \boldsymbol{\alpha})}[\log{\theta_k}] = \Psi(\alpha_k) - \Psi(\sum_{k'=1}^K \alpha_{k'})$, ...
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Expected value of the product of three random variables
For two dependent random variables we have:
$$Cov[X, Y] = E[XY] - E[X]E[Y]$$
So that $E[XY] = E[X]E[Y] + Cov[X, Y]$
In case of three arbitrarily correlated random variables $(X, Y, Z)$, is it possible ...
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Is a product that has 4.9 stars from ten customers better than one that has 4.5 stars from a hundred customers?
In many areas, we encounter a situation where we compare averages of highly skewed statistics using two unequally sized samples. Typically, this happens when comparing items in an online store. For ...
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Tight upper bound on the function of expected value
Let $R$ be a positive integer, $\mathcal{X}$ be the sample space and $x \in \mathcal{X}$ be an event of the sample space; $P(x)$ denotes the probability of occurrence of event $x$. The problem is to ...
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What conditions are there on the exponent $p$ such that $\underset{\mu}{\arg\min}\left\{\mathbb E\left\vert X-\mu\right\vert^p\right\} $ must exist?
Let $X\sim F(x)$ be a (univariate) random variable defined by distribution function $F$. If the expected value exists, it is equal to $
\mathbb E[X] = \underset{\mu}{\arg\min}\left\{\mathbb E\left\...
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Modelling Y=min(c,X) for different c
Assume I have a random variable $X \sim Poisson(\lambda)$ which models the potential nr of people entering some room. Now consider this room has a capacity $c$ so that whenever $X > c$ we observe $...
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Find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$
Truth be told, I don't really have an issue with this problem in general, but in it's calculation. Let me explain.
We need to find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$, $1<x<4$ and $y>0$...
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What is the bias of uniform distribution parameter estimator?
I have a question regarding question 2 of chapter 6 of "All of Statistics" book by Larry Wasserman.
let: $$X_1, ... , X_n \sim \operatorname{Uniform}(0, \theta )$$
and let:
$$\hat{\theta} = \...
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Expected value of decreasing function of random variable versus expected value of random variable
Given two random variables $X_1$ and $X_2$ (same sample space $\mathcal{X}$) that
$$\mathbb{E}[X_1]=\int_{\mathcal{X}}xf_1(x)dx > \mathbb{E}[X_2]=\int_{\mathcal{X}}x f_2(x)dx$$
Can we say that $\...
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How to calculate the expectancy of the ratio of non-independent random variables?
How can I calculate this expectancy:
$$
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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Correct notation for uncertain expectation
I need to write some documentation for a couple of process design options. Ultimately, my organization has to find a way to estimate the values of vector $A$.
I have come up with a model to calculate $...
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Expectation of KL-divergence only as the log ratio of the probabilities
In the DPO paper, and in particular in the proof attached below, how can we expand the KL divergence only as the log ratio of the probabilities of the two distributions?
According to the definition ...
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How to find probability from $E[X^n]$?
It is given that $E[X^n] = \frac{2}{5}(-1)^n + \frac{2^{n+1}}{5}+\frac{1}{5}$, where $n=1,2,3,\ldots.$
I need to find $P(|X-\frac{1}{2}| > 1)$.
What my approach is :
I have opened the modulus ...
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