0
votes
1answer
39 views

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% ...
1
vote
1answer
51 views

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 ...
0
votes
1answer
61 views

Given $f_X$. Integrate $\int_0^\infty \log_2 (x+1) f_X \, dx$.

Say $Y=Log_2[1+x]=g(X)$ and $f_X = \frac{e^{-\frac{(\mu -\log (x))^2}{2 \sigma ^2}}}{\sqrt{2 \pi } x \sigma }$ is Log-normal density function: [Wiki] Find E[Y]? Since $E[Y] = \int_0^\infty y f_Y \ ...
1
vote
1answer
61 views

Just learned about the bell curve in statistics. How is calculus related to this curve?

I'm learning about the bell curve in statistics and I'm trying to understand the calculus behind the concept. I've taken calc 1 already. How is the integral related to this ...
0
votes
2answers
62 views

Painful? Moment Generating Function

Part 1 Let $X$ be a random variable with the p.d.f. $f(x)=\frac{1}{4\pi}e^{\frac{-x^2}{4}}$, compute the MGF of $X$. So I know I want ...
1
vote
0answers
44 views

Is the variance of the left truncated normal distribution decreasing in lower bound?

I am wondering whether the variance of the left truncated normal distribution is always decreasing in $\alpha$ (lower bound)? The untruncated distribution of x is $\mathcal{N}(\mu,\sigma^2)$. The ...
0
votes
1answer
83 views

Differentiating mahalanobis distance

I would like to differentiate the mahalanobis distance: $$D(\textbf{x}, \boldsymbol \mu, \Sigma) = (\textbf{x}-\boldsymbol \mu)^T\Sigma^{-1}(\textbf{x}-\boldsymbol \mu)$$ where $\textbf{x} = (x_1, ...
3
votes
2answers
203 views

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 ...
9
votes
3answers
470 views

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 ...
1
vote
0answers
73 views

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 ...
2
votes
0answers
69 views

Integral of the Normal Characteristic Function

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{ ...
1
vote
2answers
104 views

Bound for erf function

For small $\epsilon \geq 0$ Is $erf(\epsilon) \leq \epsilon$ Can somebody give me the hint
1
vote
1answer
185 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} ...
2
votes
1answer
131 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 ...
1
vote
2answers
268 views

How to show that the inverse Gaussian density integrates to 1?

How to prove $\int_{0}^{\infty}\left[\frac{\lambda}{2\pi x^3}\right]^{1/2}\exp\left\{\frac{-\lambda(x-\mu)^2}{2\mu^2 x}\right\}dx=1$?
2
votes
0answers
168 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 ...
3
votes
1answer
288 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 ...
1
vote
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) = ...
0
votes
1answer
1k views

How to apply Central Limit Theorem to Uniform Distribution to generate Normal Distrubution?

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 \\ ...
1
vote
5answers
1k views

Quantile function with Normal distribution and Weibull distribution

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 ...