1
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1answer
48 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
59 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
54 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
61 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
41 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
79 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
183 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
454 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
72 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
67 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
99 views

Bound for erf function

For small $\epsilon \geq 0$ Is $erf(\epsilon) \leq \epsilon$ Can somebody give me the hint
1
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1answer
181 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
127 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
257 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
167 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
276 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
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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 ...