# Tagged Questions

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### Determine the target weight so that no more than 5% of boxes with normal weight distribution contain less than 500 g [closed]

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. Suppose a law states that no more than 5% ...
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### How to fill in these steps to evaluate this Gaussian integral?

As a part of a much bigger problem, I came across this integral $$\int_{-\infty}^{\infty}\ln(|x|)\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}dx$$ which represents ...
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### Taking a derivative with respect to a matrix

I'm studying about EM-algorithm and on one point in my reference the author is taking a derivative of a function with respect to a matrix. Could someone explain how does one take the derivative of a ...
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### Is the mean of the truncated normal distribution monotone in $\mu$?

I am wondering whether the mean of the truncated normal distribution is always increasing in $\mu$. The untruncated distribution of $x$ is $\mathcal{N}(\mu,\sigma^2)$. The mean of the truncated ...
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### Integrals of derivatives of normal distribution multiplied by polynomial?

Is there anything in the literature related to obtaining bounds of integrals of the form: $$\int_{\mathbb{R}} |P^{(k)}(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ where $P(t,z)$ the density of ...
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The characteristic function of the $N$-variate Normal distribution is \forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ ... 2answers 111 views ### Bound for erf function For small \epsilon \geq 0 Is erf(\epsilon) \leq \epsilon Can somebody give me the hint 1answer 196 views ### Normal Distribution Identity I have the following problem. I am reading the paper which uses this identity for a proof, but I can't see why or how to prove its true. Can you help me? \begin{align} \int_{x_{0}}^{\infty} e^{tx} ... 1answer 137 views ### Techniques for evaluating probability integral Consider the integral of a normal distribution:\int_a^b f(x)\,\mathrm d x=c $$and a second integral for the expected value:$$ \int_a^b x\cdot f(x)\,\mathrm dx $$Since you know the first ... 2answers 284 views ### How to show that the inverse Gaussian density integrates to 1? How to prove \int_{0}^{\infty}\left[\frac{\lambda}{2\pi x^3}\right]^{1/2}\exp\left\{\frac{-\lambda(x-\mu)^2}{2\mu^2 x}\right\}dx=1? 0answers 169 views ### Proof of a gaussian integral turning into a cosine I have a numerical evidence of$$\int_0^{1/2} \frac{1}{\sqrt{2\pi}\sigma_0x}\exp\left(-\frac{(\mu_0x-y)^2}{2\sigma_0^2x^2}\right)dx \approx 1+\cos(2\pi y),$$where ... 1answer 300 views ### Is there a closed-form expression for the integral of this product of gaussian functions? Considering:$$f(x) = \frac{1}{\sigma_x\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x}{\sigma_x})^2}g_i(x) = \frac{1}{\sigma_i\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{a_i+b_ix}{\sigma_i})^2}$$Is there a ... 1answer 3k views ### Linear transformation of normal distribution Not sure if "linear transformation" is the correct terminology, but... Let X be a random variable with a normal distribution f(x) with mean \mu_{X} and standard deviation \sigma_{X}:$$f(x) = ...
Suppose I have a simple uniform continuous "unit" distribution X: \begin{align*} \forall y \in \mathbb{R} \implies \\ y < 0 : & P(X < y) = 0 \\ y \in [0,1] : & P(X < y) = y \\ ...
A quantile function Q is defined in terms of its distribution function F as: $Q(p)=inf\{ x\in R:p \le F(x)\},p\in(0,1)$ But i don't understand very well how it works exactly. Suppose we are managing ...