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Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?

This is another illustration of Jensen's inequality $$\mathbb E[\log X] < \log \mathbb E[X]$$ (since the function $x\mapsto \log(x)$ is strictly concave] and of the more general (anti-)property ...
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Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?

Consider two values symmetrically placed around $0.5$ - like $0.4$ and $0.6$ or $0.25$ and $0.75$. Their logs are not symmetric around $\log(0.5)$. $\log(0.5-\epsilon)$ is further from $\log(0.5)$ ...
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How to show that this integral of the normal distribution is finite?

Intuitively, the result is obvious because (a) $\phi$ is a rapidly decreasing function (its magnitude decreases at a quadratic exponential rate) and (b) $\Phi$ is bounded above and, for negative $x,$ ...
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• 107k
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• 5,456

Expected loss function from bias variance trade off (integral help)

Starting from the objective $$\min_{\hat f}\mathbb E_D\left[\iint (y-\hat f(x))^2 p(x,y)\,\text dx\text dy \right]$$ the derivation of the optimal function$$\hat f: x\longmapsto \hat f(x)$$follows ...
• 107k
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Compute median of continuous distribution using integrate() in R

CAUTION! As was pointed out and explained by whuber in the comments, the current code below does not check if it is fed a density that integrates to one (or to some other finite value which we could ...
• 34.2k
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How to deduce $\mathbb{E}(\sqrt{X}) < \infty \implies\int_{\mathbb{R}^+} (1 - F(x))^2 dx < \infty,~X$ being a non-negative integrable rv?

Interesting problem. The proof goes as follows: let $X, X_1, X_2$ i.i.d. $\sim F$, then \begin{align} \int_0^\infty (1 - F(x))^2dx = E[\min(X_1, X_2)] \leq E[\sqrt{X_1}\sqrt{X_2}] = (E[\sqrt{X}])^2 ...
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Variance of the integral of a stochastic process

Interchanging the order of integration and expectation you get $$E(I)=E\int_0^L X(t) dt = \int_0^L EX(t) dt = \int_0^L \mu dt = L\mu$$ and similarly, the second moment of $I$ becomes \begin{align} E(...
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Expressing a marginal probability using copulas

The issue of notation seems crucial. I propose, therefore, to disambiguate the ubiquitous and overloaded "$f$" by means of subscripts. Thus, $f_{XYZ}$ will be the full density function and ...
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Plain English explanation of Ito's integral?

I think you're confusing the Lebesgue integral with Itô calculus. They are related concepts. I'll explain. Lebesgue vs Riemann The simplest explanation of the difference between Lebesgue and Riemann ...
• 61.9k
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Plain English explanation of Ito's integral?

Ito's integral has nothing to do with the area under a curve and no connection with rectangles, random or otherwise. Let me try to share a motivation in the simplest language I can manage. There's ...
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Monte Carlo simulations for arbitrary functions

My purpose here is to show a Riemann approximation for $\int_0^1 x^2\, dx = 1/3$ with enough rectangles to approximate the integral. Then to do a Monte Carlo integration in which uniformly chosen ...
• 57k
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