3
votes
2answers
62 views

$\frac{1}{\sqrt{2\pi}}\int_\frac {1}{2}^0\exp(-x^2/2)dx$

How do we analytically evaluate $J=\frac{1}{\sqrt{2\pi}}\int_\frac {-1}{2}^0\exp(-x^2/2)dx$? This is what I tried: $$ J^2=\frac{1}{{2\pi}}\int_\frac {-1}{2}^0\int_\frac {-1}{2}^0\exp(-(x^2+y^2)/2)dxdy ...
1
vote
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 ...
3
votes
0answers
43 views

On integration of a Gaussian-like function over a region $g(\mathbf{x})\leq 1$

Let $X$ be a random variable which follows an $n$-dimensional Gaussian distribution with mean vector $\mu\in\mathbb{R}^n$ and covariance matrix the symmetric positive definite $n\times n$ matrix ...
0
votes
1answer
59 views

integral with pdf of a gaussian

$$ I = \int_{0}^{\infty} x \phi(x) dx $$ where $\phi(x)$ is the pdf of a normal distribution. Here I read that: If $X = \mu + \sigma U$ with $U$ a std normal, $$ I = E[\mu + \sigma U; mu + \sigma ...
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 \ ...
0
votes
1answer
27 views

How to calculate $\int\limits_{-\infty}^{+\infty} \frac{n-1}{\alpha}\Phi(\frac{x}{\alpha})^{n-2}\phi(\frac{x}{\alpha})^2dx$?

I was working on a research project that involves taking the integral of $$\frac{n-1}{\alpha}\int\limits_{-\infty}^{+\infty} ...
1
vote
1answer
15 views

Integrating the error function in a calculation related to Brownian motion

I wish to calculate the probability that a standard linear Brownian motion $B(t)$, $t\ge 0$, will be at time $t_0$ inside the interval $[a,b]$, and at time $t_1$ in the interval $[c,\infty)$. To do ...
2
votes
1answer
60 views

Upper bound of difference of squares of quantile standard normal

Let $\Phi$ denotes the cummulative standard normal distribution and $\Phi^{-1}$ denotes its inverse. Given $u,v\in[0,1)$. I'am going to find an upper bound of $$ ...
2
votes
2answers
83 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
1
vote
0answers
41 views

How to simplify the computation of a special case of multivariate normal cdf

I am trying to compute a multivariate normal cdf where all but the last bounds of the integrals are symmetric: $$F(a, \sigma, m ) = ...
1
vote
1answer
27 views

Integral of cumulative normal

Let $$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} \exp\left({-\dfrac{\omega^2}{2}}\right) d\omega.$$ Question: for what values of $a$, $b$ and for what choices of $f(x)$ would the following ...
1
vote
1answer
41 views

Integration involving complicated exponential form

I'm trying to simplify the following: $\int_0^ts^{-\frac{3}{2}}e^{-\frac{(a+bs)^2}{2s}}~ds$ Basic substitution always gives a $s^{-\frac{1}{2}}~ds$ counterpart which I don't know how to get rid of. ...
1
vote
1answer
55 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
0
votes
0answers
13 views

Expectation of a function of multivariate normal cdf

Can someone help me find the following expectation $E_Y(Y*\Phi_k(a+BY|\eta,\Omega))$ where $Y \sim N_n(\mu,\Sigma)$ ? I know that $E_Y(\Phi_k(a+BY|\eta,\Omega))=\Phi_k(a|\eta-B\mu,\Omega+B\Sigma ...
0
votes
0answers
41 views

Efficient approximation of derivatives of an integral

Suppose $ \phi(z) $ is the probit function (http://en.wikipedia.org/wiki/Probit). And $$ Z = \int \phi(\mathbf{w}^\top \mathbf{x}) \mathcal{N}(\mathbf{w}; \mathbf{\mu}, \mathbf{\Sigma}) d\mathbf{w} ...
2
votes
0answers
91 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
0
votes
0answers
44 views

Expected Value Question (normal Distribution)

I'm trying to calculate $E(X)$ where $f(x)$ is a variable such that; f(x) = 0 , -infinity<=x $$f(x)= \begin{cases} 0 \ , &-\infty \le x \lt c_1\\ x-c_1 \ , & c_1 \le x \lt b \\ b\ , ...
2
votes
1answer
20 views

Integral arising from Brownian motion question

I want to show that $\int_0^{\infty}exp({-a^2 / {2t} - \lambda t})\frac{a}{\sqrt{2\pi t^3}} dt = exp(-a \sqrt{2 \lambda})$. Please can you give me a clue on how to do this. I have tried integration by ...
0
votes
0answers
22 views

Multivariate normal distribution of circular object

my Problem is the following: I have a circular object that is moving around. I also have the covariance matrix for the position of the object $(x,y)$. So far, I used the multivariate normal ...
1
vote
0answers
211 views

Integral of a random process that follows Gaussian Process

Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$ How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that ...
0
votes
1answer
57 views

normal distribution expected value

In this derivation: http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html $$f(x) = \sqrt{\frac{k}{2\pi}}e^{-\frac{k(x-\mu)^2}{2}}$$ they let $x-\mu = v$ $dx = dv$ and conclude that ...
2
votes
2answers
171 views

Conditional mean and variance of normal random variables

There are two independent normal random variables $N_1, N_2$ with means $\mu_1, \mu_2$ and variances $\sigma_1^2, \sigma_2^2$ respectively. Is there a way to compute the two conditional expressions ...
2
votes
0answers
94 views

How to integrate the following formula about normal distribution

How to compute the following formula? $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, dx $$ $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, xdx $$ where ...
1
vote
1answer
171 views

normal distribution derivation

In this derivation: http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html how do these equal? $$ -k\int (x-\mu) dx = -\frac{k}{2} (x-\mu)^2$$ Isn't this the case? $$ -k\int (x-\mu) dx ...
-1
votes
2answers
88 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: ...
3
votes
2answers
321 views

How to approximate the integral of the standard normal distribution.

So I have this eqn. $$ f(x)= \frac {e^ \frac{-x^2}{2}} {\sqrt{2\pi}} $$ I need to find: $$ \int\limits_{-1}^1 f(x)dx $$ So I want to use this series to integrate. I know that: $$ e^x = ...
0
votes
1answer
104 views

integrate moments normal distribution between finite limits

Can somebody help me to evaluate the following integral: $$\frac{1}{\sqrt{2\pi}\sigma}\int_a^b x^2 \exp\left(\frac{-x^2}{2\sigma^2}\right)\mathrm dx$$ Answer involving cumulative normal (erf) would ...
2
votes
1answer
113 views

Gaussian function

I want to scale the Gaussian function $\exp(-x^2)$ to the unit disc. In particular, I wish to represent $\int_0^\infty \exp(-x^2) dx$ as $\int_0^1 g(x) dx$, where $g$ should be the rescaled Gaussian ...
7
votes
3answers
2k views

Derivation of the density function of student t-distribution from this big integral.

My lecturer posed a question where we derive the density function of the student t-distribution from the Chi-square and Standard normal distribution. I worked on this question for days, and I am ...
1
vote
0answers
26 views

Confusion related to predictive distribution of gaussian processes

I have this confusion related to the predictive distribution of gaussian process I didn't get how the integration gave that result. What is P(u*|x*,u). Also how come the covariance of the posterior ...
3
votes
1answer
182 views

Help with gaussian integral

I need to solve this gaussian integral: $$\int_\mathbb{R} (2\pi)^{-n/2}\mid \Sigma\mid ^{-\frac{1}{2}}e^{-\frac{1}{2}(u-Kx)^T\Sigma ^{-1}(u-Kx)} u^TRu \,\mathrm du$$ It is the integral of a ...
2
votes
3answers
186 views

How to integrate the difference between the CDFs of two normal distributions

I have two normal distributions A and B. I am trying to write a program that will take mean(A), stddev(A), mean(B), stddev(B) and output the result of the following equation: $$ ...
8
votes
2answers
220 views

Computing the Gaussian integral with step functions

Say, we are interested in deriving $$\int_{-\infty}^{\infty}e^{-x^2}=\sqrt{\pi}\tag{1}$$ There are many well known ways to do it, for example: by polar coordinates via the gamma function, etc. ...
1
vote
1answer
28 views

Changing bounds of integrals

If I have: $$\int_{L}^{\infty }e^{\dfrac{-(x-\sigma \sqrt{T})^2}{2}}\,dx$$ let $y = x - \sigma \sqrt{T}$ $$\int_{L - \sigma \sqrt{T}}^{\infty }e^{\dfrac{-y^2}{2}} \, dy$$ Why does the lower bound ...
1
vote
1answer
107 views

Integral involving normal densities

I am trying to solve the integral $$I(y)=\int_{\mathbb R}f(x,y)g(x)dx,$$ where $f(x,y)$ is the bivariate normal density with known mean $(\mu_1,\mu_2)$ and covariance matrix $\Sigma$ , and $g(x)$ is ...
1
vote
1answer
80 views

integral of normal distribution

how to do this integral: $$ \mathop{\int\int}_{y+2x>0} x y \frac1{2\pi\sigma_x\sigma_y}e^{ -\frac{(x-\mu_x)^2}{2\sigma_x^2}}\cdot e^{ -\frac{(y-\mu_y)^2}{2\sigma_y^2}} dx dy$$ Both x and y are ...
0
votes
1answer
118 views

how to do this integral: $ \int_{0}^{\infty} \int_{0}^{\infty} x y \phi(x, y) dx dy$

how to do this integral: $$ \int_{0}^{\infty} \int_{0}^{\infty} x y \phi(x, y) dx dy$$ where $ \phi(x,y)$ is a general pdf of bivariate normal distribution, that is: $$\phi(x,y) = ...
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
0
votes
1answer
46 views

Bound for the integral

Is there any way to bound the following integral $$\int_{-(\epsilon+1)/\sigma}^{(\epsilon-1)/\sigma} \mathrm e^{-t^2/2}\,dt$$
1
vote
0answers
217 views

Bayesian posterior with integrals over normal densities

Realizations from normal distributions with known precision are used to estimate the mean, but the realizations are not always precisely observed. Instead, only a range of the realization is observed. ...
2
votes
1answer
82 views

Dirac function and integration by parts

I have some problems to show the following relation, apparently using integration by parts and knowing that $\phi$ denotes the density of the standard one dimensional normal distribution. $$\int ...
0
votes
1answer
111 views

How can I solve this integral?

How can I solve the following integral? $$\int_{-\infty}^\infty \prod_{i=1}^n \bigg( 1 - \Phi\left(\frac{c - \mu_i}{\sigma_i}\right) \bigg) \frac{1}{\sigma_Y}\phi \bigg(\frac{c-\mu_Y}{\sigma_Y} ...
0
votes
0answers
421 views

Numerical integration of 2-d Gaussian Distribution in MATLAB

I am looking for a really fast way to integrate numerically the 2-dimensional gaussian density with identity covariance matrix ...
0
votes
3answers
500 views

Integrating the pdf of a normal distribution

I need to find the distribution of $Y=X_1+X_2$ where both $X_1$ and $X_2$ are normally distributed with $(\mu,\sigma^2)$. So I'm looking for ...
2
votes
1answer
141 views

Conditional Expectations (Mainly an integral question)

Let $X_1$ and $X_2$ be two Random variables with a standard normal distribution, and the two variables are independent. Find $E[X_1|X_1>X_2]$ My answer is far. If we knew $X_2$, then the answer ...
4
votes
1answer
254 views

Confusion related to integral of a Gaussian

I am a bit confused about calculating the integral of a Gaussian $$\int_{-\infty}^{\infty}e^{-x^{2}+bx+c}\:dx=\sqrt{\pi}e^{\frac{b^{2}}{4}+c}$$ Given above is the integral of a Gaussian. The ...
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 ...
5
votes
3answers
535 views

Compute probability of a particular ordering of normal random variables

There are $m$ normally distributed, independent random variables $N_1, \ldots, N_m$ with distinct means $\mu_1, \ldots \mu_m$ and standard deviations $\sigma_1, \ldots, \sigma_m$. Then, we get a ...
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 ...