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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Expected value of a decreasing function of two random variables
My question is exactly equal to the question posted at Expected value of decreasing function of random variable versus expected value of random variable with just one extra assumption: the two random ...
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Bayesian Fubini Tonelli
I am working on a bayesian framework where I place a Gaussian Process on my function $f\sim GP$ and have data $D^n=\{(X_i,Z_i,W_i)\}^n$.
I then have the posterior measure $\mu(f|D^n)$. The posterior ...
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Validating binary prediction model
Suppose we have a model that predicts for binary event $e$ ($0$ or $1$) with a single output $p$ (the expected probability $e$ occurs).
If we are able to compare $p$ with the true value of $e$ ($0$ or ...
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Deriving home vs away goals - using total expected goals and home/draw/away probabilities
In the context of a football ("soccer") match, if I have the following for a single game:
Probability of Team A winning
Probability of Team B winning
Probability of a draw
The total goals ...
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$E[(X+Y)^{a}] > E[(X)^{a}]$?
Assume I have two strictly positive i.i.d. random variables, $X$ and $Y$. Under what conditions is the following inequality true?
$$E[(X+Y)^{a}] > E[(X)^{a}], \hspace{2mm} a \in (0,1)$$
Should have ...
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Probability algorithm on strings
Let $x$ be any binary string $\in (0,1)^*.$
The majority language is given by:
$$\text{MAJ}:=\{x\in (0,1)^*:\sum_{i=1}^ {|x|}x_i>\frac{|x|}{2}\},\text{where $x_i$ is the $i$-th position value(...
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Need help in calculating $\mathbb{E}(\frac{1}{x_{(2)}-x_{(1)}}\int_{x_{(1)}}^{x_{(2)}} f(t) \ dt)$, where $x_{(i)}$ are related Beta distribution
Suppose $Y, Z \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$.
Let $a = g(\min(y,z)),\ b=g(\max(y,z)).$
How can I calculate the expectation $$\mathbb{E}\left[\frac{1}{b-a}\int_a^b f(t) \ dt\right]$$ ...
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Derive the expectation and variance of squared sample correlation: delta-method or else?
I would like to obtain the expectation and variance of the squared Pearson sample correlation ($\operatorname{E}(R_{lk}^2)$ and $V(R_{lk}^2)$) between two random variables $l$ and $k$ following a ...
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Why does $E(V_n/(n+2)-1)^2=2/(n+2)$ when $V_n\sim\chi^2(n)$?
I was reading some lecture notes when I saw a simplification I didn't understand. Specifically, we have $V_n\sim\chi^2(n)$. It was then written then
$$E\left(\frac{1}{n+2}V_n-1\right)^2=\frac{2}{n+2}.$...
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Expectation of $u^\top(u+Ax)$, when $A$ and $u$ are nonlinear functions of $x$
Let $x\in\mathbb R^d$, and $s=\operatorname{softmax}(x)$. Let $y$ be a fixed one-hot vector such that
$$u = s-y \\
v =(\operatorname{diag}(s) - ss^\top)x$$
I am interested in the inequality $u^\top (u ...
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Expected value of Cosinus in High dimension
I would like to prove that the cosinus of the angle formed by 3 randomly points tends to $\frac{1}{2}$ as the dimensionality tends to $\infty$. Could it be solved with the expected value formula ? It ...
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Expectation over cost-normalized Expected improvements
Are the following two expressions equivalent if we assume the independence of f(x) and C(x)?
$$
E\left[\frac{E\left[\max\left(f(x) - f(x^*), 0\right)\right]} {C(x)}\right]
$$
$$
\frac{E\left[\max\...
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Expectation & Covariance matrix of indicator vector
Suppose we have the $p$-dimensional random vector $\boldsymbol{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$. Take the set $A$ to be (without loss of generality) the negative real line, thus $A = (- \...
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How to calculate the expectation of the following Dirichlet distribution and Beta distribution?
This is a question from my research, related to the derivation of the variational EM algorithm with mean-field assumption about LDA-based model.
We all know, given that $\boldsymbol{\theta} \sim \...
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Verifying the integrability condition of a deterministic volatility function
Suppose there is integrability condition:
\begin{equation}
\mathbb{E}\left[\int_0^T\frac{\sigma^2(t)}{T-t}dt\right]<\infty
\end{equation}
for an arbitrary volatility function. Suppose I nominate ...