# Tag Info

### The variance of the resulting Gaussian is found by taking the second derivative of the exponent?

I've read the paper recently and wondered the same thing. The Cramér–Rao bound (see this and this) states that the variance of an unbiased estimator cannot be lower than the inverse Fisher information,...
1 vote
Accepted

### How to find the power function given exponential distribution?

You are right. There is an error. The upper limit of integration must be $0.05$. The power is defined by the formula $\inf_{\theta \in \Theta_1} \mathbb{E}_{X \sim \theta}[T(X)]$ for any test $T$. In ...
• 2,410

### Questions on Box and whiskers diagram

Heropup wrote a pretty good answer, but I wanted to add an answer under the assumption that all the data points would be evenly distributed within their respective quartiles. Since each box and each ...
• 184
1 vote
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### Sub-gaussian norm

For any centered random variable $X$, let $\Sigma(X) = \{\sigma \geq 0 : X \text{ is } \sigma \text{ sub-Gaussian} \}$. Thus, $\|X\|_{vp} = \inf \Sigma(X)$. Let us begin with positive definiteness. ...
• 2,796

### Evaluating $\lim_{n\to\infty}\sum_{i=1}^n \exp\left(\frac{-|x-X_i|^2}{2(\sigma/n)^2}\right)$

Thanks to some helpful suggestions, I was able to find a solution. For simplicity, I will prove it for the case of $\mathbb{R}^d$ and when $\rho$ is a K-Lipschitz continuous density. Define the ...
• 136

### Questions on Box and whiskers diagram

Skewness is difficult to discern from a box-and-whisker plot, so I would not assume that in general one can assert extent of skew from such diagrams. In cases where we can make assumptions about the ...
• 139k
1 vote

### concentration of maximum of gaussians

For the sake of completeness, I will prove here the inequality $$P(\|X\|_\infty \geq \sqrt{2 \log (2n)}+t)\leq \frac{1}{2}\exp(-t^2 /2) \quad (t>0),$$ which is slightly ...
• 3,638
Accepted

### Method of Moment for Normal mixtures $p\cdot N(0, 1) + q\cdot N(\eta, 1)$

If $Y_0 \sim N(0,1)$ and $Y_\eta \sim N(\eta,1)$ and the $X_i$ are iid with the mixture distribution (a probability $p$ of following $Y_0$'s distribution and a probability $q$ of following $Y_\eta$'s ...
• 158k

### On expectation of maximum of gaussians

For the sake of completeness, I give here the full proof of the existence of $M>0$ such that $\mathbb{E} \left[ \max_{i}|X_i|\right] \leq M \mathbb{E} \left[ \max_{i}X_i\right]$ for all $n \geq 2$ (...
• 3,638
Accepted

### If $X\sim N(\mu, \sigma^2)$ and $\Phi$ is the CDF of a standard Normal random variable, what is the distribution of $\Phi(X)$?

Since $\ \Phi\$ is strictly increasing, it has a strictly increasing inverse $\ \Phi^{-1}:(0,1)\rightarrow(-\infty,\infty)\ .$ Then, for $\ 0<x<1\ ,$ \begin{align} P(\Phi(X)\le x)&=P\big(X\...
• 29.3k

• 2,683
1 vote
Accepted

• 884
1 vote
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### The gap distribution of the random variables.

Let $F(x)$ be the CDF of your continuous random variable $X_j$ and $f(x)$ its density; you have $n$ of them and $i\in\{2,3,\dots,n\}$. For given two real numbers $a<b$, the probability that $k:=i-1$...
• 2,961

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