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 softmax transformation of Gaussian random vector
Let $\mathbf w_1,\mathbf w_2,\ldots,\mathbf w_n \in \mathbb R^p$ and $\mathbf v \in \mathbb R^n$ be fixed vectors, and $\mathbf x \sim \mathcal N_p(\boldsymbol{\mu}, \mathbf{\Sigma})$ be an $p$-...
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Concentration of maximum of subexponential random variables
I'm looking for a concentration bound on the maximum of a collection of sub-exponential random variables, which are not necessarily independent. More specifically, I have the following collection:
\...
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Expected value of a "logistic uniform" multivariate
Let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb R^d$ and $b_1,\ldots,b_n \in \mathbb R$ be fixed. For $\mathbf{x} \sim \mathcal U([0,1]^d)$ and $j \in \{1,\ldots,n\}$, consider the real variable ...
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Assign Numbers to N-sided Die to Make the Expectation Close to the Fair Mean
I got the following question in one interview: suppose we have an N-sided die and given the probability of landing on each side, how to assign values from 1 to N, to make the expected value close to $(...
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Weighted Conditional Expectation definition in AdaBoost
I am looking at "Additive logistic regression a statistical view of boosting" paper (https://web.stanford.edu/~hastie/Papers/AdditiveLogisticRegression/alr.pdf)
In page 346, the authors ...
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Finding the Cauchy Principal Value Mean of the pdf for $Z=XY$ where $\mathbb{E}[Y]$ does not exist
Let's first define the Cauchy Principal Value Mean (PVM). For a continuous random variable $V$ with pdf $f_{V}(v)$, the PVM of $f_{V}$ is
\begin{equation}
\mathrm{PV}(\mathbb{E}[V])=\lim_{a\to\infty}\...
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Expectation of a strictly increasing function
Assume that $X_1$ and $X_2$ are two i.i.d. random variables with pdf $f$. Also, assume that $a$ and $b$ are two fixed real numbers such that $a>b$. If $g$ is a strictly increasing function, do I ...
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Expected number of multinomial samples to cover a multiset
Consider a multinomial distribution $[p_1, \ldots, p_n]$ and a collection of counts $[a_1, \ldots, a_n]$. I would like to know the expected number of multinomial samples needed until every element $i$ ...
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Does the law of total probability apply to hazards?
Consider the hazard function for a random variable $T$, conditional on some other random variable $U$:
$$
h(t|U=u)=\lim_{\Delta t\rightarrow0}\frac{P(t<T<t+\Delta t|T>t,U=u)}{\Delta t}
$$
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The expected value of $\frac{1}{\sqrt{1-r}}$ where $r$ is Pearson correlation
I am looking to unbias the sample statistic $\frac{1}{\sqrt{1-r}}$ where $r$ is a Pearson correlation. The population is assumued binormal with equal variance $\sigma$ and with true correlation $\rho$....
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Calculating the variance of a weight-average when the weights also have a variance
Assume there is a series of random variables $X_1$, $X_2$, ..., $X_N$ representing a series of values to be weight-averaged, and a corresponding series of random variables $W_1$, $W_2$, ..., $W_N$ ...
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Expectation of a random walk that can't go below zero
Suppose we have a random walk $S_n$ that is constrained to be positive or zero, that is:
$$S_0 > 0$$
$$S_{i+1} = \max(S_i+x_i,\space 0)$$
$$x_i \sim N[\mu,\sigma^2]$$
Can we analytically ...
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Is the expected value a valid decision-making method in a very short term?
This might be related to game theory more than statistics, but I decided to ask this question here.
Let's assume you're offered a lottery. There are a hundred balls in a bowl: 99 white balls and one ...
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What do you think of this proof for Fisher information?
I want to prove This formula:
The score function is basically the derivative of the maximum likelihood's log, so to get the information I make another derivative of that:
$$ -E[∂/∂θ s(X;θ)] = -E[∂/∂θ ...
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What is the expected value of $X_i/\|X\|^2$ when $X \sim \mathcal{N}(\mu, \sigma^2I)$
Let $X$ be an N-dimensional normal random vector with non-zero mean $\mu$ and diagonal covariance matrix $\sigma^2I$. I would like to understand if it is possible to derive the expected value of the ...
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