# Tagged Questions

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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 $E[Y] = \int_0^\infty y f_Y \ ... 1answer 27 views ### How to calculate$\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 $$\frac{n-1}{\alpha}\int\limits_{-\infty}^{+\infty} ... 1answer 15 views ### Integrating the error function in a calculation related to Brownian motion I wish to calculate the probability that a standard linear Brownian motion B(t), t\ge 0, will be at time t_0 inside the interval [a,b], and at time t_1 in the interval [c,\infty). To do ... 1answer 60 views ### Upper bound of difference of squares of quantile standard normal Let \Phi denotes the cummulative standard normal distribution and \Phi^{-1} denotes its inverse. Given u,v\in[0,1). I'am going to find an upper bound of$$ ... 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 ... 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: $$F(a, \sigma, m ) = ... 1answer 27 views ### Integral of cumulative normal Let$$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} \exp\left({-\dfrac{\omega^2}{2}}\right) d\omega.$$Question: for what values of a, b and for what choices of f(x) would the following ... 1answer 41 views ### Integration involving complicated exponential form I'm trying to simplify the following: \int_0^ts^{-\frac{3}{2}}e^{-\frac{(a+bs)^2}{2s}}~ds Basic substitution always gives a s^{-\frac{1}{2}}~ds counterpart which I don't know how to get rid of. ... 1answer 55 views ### Integration of standard multivariate normal distribution We should express the integral I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x using I_1. Where \left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ... 0answers 13 views ### Expectation of a function of multivariate normal cdf Can someone help me find the following expectation E_Y(Y*\Phi_k(a+BY|\eta,\Omega)) where Y \sim N_n(\mu,\Sigma) ? I know that E_Y(\Phi_k(a+BY|\eta,\Omega))=\Phi_k(a|\eta-B\mu,\Omega+B\Sigma ... 0answers 41 views ### Efficient approximation of derivatives of an integral Suppose \phi(z) is the probit function (http://en.wikipedia.org/wiki/Probit). And$$ Z = \int \phi(\mathbf{w}^\top \mathbf{x}) \mathcal{N}(\mathbf{w}; \mathbf{\mu}, \mathbf{\Sigma}) d\mathbf{w} ... 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 ... 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 $$f(x)= \begin{cases} 0 \ , &-\infty \le x \lt c_1\\ x-c_1 \ , & c_1 \le x \lt b \\ b\ , ... 1answer 20 views ### Integral arising from Brownian motion question I want to show that \int_0^{\infty}exp({-a^2 / {2t} - \lambda t})\frac{a}{\sqrt{2\pi t^3}} dt = exp(-a \sqrt{2 \lambda}). Please can you give me a clue on how to do this. I have tried integration by ... 0answers 21 views ### Multivariate normal distribution of circular object my Problem is the following: I have a circular object that is moving around. I also have the covariance matrix for the position of the object (x,y). So far, I used the multivariate normal ... 0answers 211 views ### Integral of a random process that follows Gaussian Process Suppose X(t) follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be \mu and autovariance \sigma^2 How to prove that \int_{0}^{T}{{X(t)}dt} is random variable that ... 1answer 57 views ### normal distribution expected value In this derivation: http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html$$f(x) = \sqrt{\frac{k}{2\pi}}e^{-\frac{k(x-\mu)^2}{2}}$$they let x-\mu = v dx = dv and conclude that ... 2answers 171 views ### Conditional mean and variance of normal random variables There are two independent normal random variables N_1, N_2 with means \mu_1, \mu_2 and variances \sigma_1^2, \sigma_2^2 respectively. Is there a way to compute the two conditional expressions ... 0answers 94 views ### How to integrate the following formula about normal distribution How to compute the following formula?$$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, dx  \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, xdx $$where ... 1answer 171 views ### normal distribution derivation In this derivation: http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html how do these equal?$$ -k\int (x-\mu) dx = -\frac{k}{2} (x-\mu)^2$$Isn't this the case?$$ -k\int (x-\mu) dx ... 2answers 88 views ### Integral of an integral with variable limits I'd like to prove the following but not sure where to start: ... 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: $$e^x = ... 1answer 104 views ### integrate moments normal distribution between finite limits Can somebody help me to evaluate the following integral:$$\frac{1}{\sqrt{2\pi}\sigma}\int_a^b x^2 \exp\left(\frac{-x^2}{2\sigma^2}\right)\mathrm dx$$Answer involving cumulative normal (erf) would ... 1answer 113 views ### Gaussian function I want to scale the Gaussian function \exp(-x^2) to the unit disc. In particular, I wish to represent \int_0^\infty \exp(-x^2) dx as \int_0^1 g(x) dx, where g should be the rescaled Gaussian ... 3answers 2k views ### Derivation of the density function of student t-distribution from this big integral. My lecturer posed a question where we derive the density function of the student t-distribution from the Chi-square and Standard normal distribution. I worked on this question for days, and I am ... 0answers 26 views ### Confusion related to predictive distribution of gaussian processes I have this confusion related to the predictive distribution of gaussian process I didn't get how the integration gave that result. What is P(u*|x*,u). Also how come the covariance of the posterior ... 1answer 182 views ### Help with gaussian integral I need to solve this gaussian integral:$$\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$$It is the integral of a ... 3answers 186 views ### How to integrate the difference between the CDFs of two normal distributions I have two normal distributions A and B. I am trying to write a program that will take mean(A), stddev(A), mean(B), stddev(B) and output the result of the following equation:$$ ... 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. ... 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 ... 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 ... 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 ... 1answer 117 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: $$\phi(x,y) = ... 0answers 67 views ### Integral of the Normal Characteristic Function The characteristic function of the N-variate Normal distribution is$$\forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ ... 2answers 99 views ### Bound for erf function For small$\epsilon \geq 0$Is$erf(\epsilon) \leq \epsilon$Can somebody give me the hint 1answer 46 views ### Bound for the integral Is there any way to bound the following integral $$\int_{-(\epsilon+1)/\sigma}^{(\epsilon-1)/\sigma} \mathrm e^{-t^2/2}\,dt$$ 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. ... 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. $$\int ... 1answer 111 views ### How can I solve this integral? How can I solve the following integral?$$\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} ... 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 ... 3answers 499 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 ... 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 ... 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 ... 1answer 126 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 ... 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 ...
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 ...