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# Questions tagged [expectation]

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### Methods for high-dimensional integration of joint PDFs over unbounded regions

I am currently grappling with an issue related to high-dimensional integration of joint Probability Density Functions (PDFs) over unbounded regions. The specifics of the problem are a bit intricate, ...
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### On the mean value taken by Bernoulli random variables with joint distribution constraints

We are given a vector $n$-dimensional random vector $\mathbf{X}$ whose components are the Bernoulli random variables $X_1, X_2, \ldots X_n$, such that the probability $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...
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### Approximation for an expectation expression

Let $\mathbf{x} \in \mathbb{C}^M$ is an unknown distributed random vector (certainly not gaussian), and matrix $\mathbf{A}\in \mathbb{C}^{M \times M}$ which is fix (known). Also, assume we know the ...
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### can we get a family of classifiers $\left\{f_n\right\}_{n \in N}$such that $\lim_{n->∞} (E_{(X_1, Y_1), ...,(X_n, Y_n) \sim \rho}[R(f_n)]-R(f_B))=0$

For a given classifier $f: \mathbb{R}^d \mapsto\{0,1,2\}$, let $$R(f):=\mathbb{E}_{(X, Y) \sim \rho}\left[\mathbb{1}_{f(X) \neq Y}\right]$$ $f_B$ the Bayes classifier. can we get a family of ...
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### How to prove emprical risk converges to expectation risk as $n\to \infty$?

For example, for a classical binary classification: $x \in \mathbb{R}^d$ and $y \in\{0,1\}$ let empirical risk be $R_{\ell}^n(f):=\frac{1}{n} \sum_{i=1}^n \ell\left(f\left(X_i\right), Y_i\right)$ and ...
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### On average, how many uniformly random real numbers $u$ are needed for their sum to exceed $1$, if $u_1$ is in $(0,1)$ and $u_k$ is in $(0,eu_{k-1})$?

A well-known question is: on average, how many uniformly random real numbers in $(0,1)$ are needed for their sum to exceed $1$? The answer is $e$. Let's tweak this question by making each random ...
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### expectation of log(1-x^a) if x is a beta random variable

How can I compute $\mathbb{E}_{q}\Big[\log (1-x^a)\Big]$ when the distribution of $q$ is given as $q(x)\sim\mathrm{Beta}(\alpha,\beta)$?
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### Expectation of period length of functions $f:\{1,\ldots,n\}\to \{1,\ldots,n\}$
For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ associate a sequence $\text{seq}(f))$ defined recursively by ...