3
votes
1answer
57 views

Integral of 2D Gaussian over triangle

I am interested in computing (numerically) the integral of a 2D Gaussian (with arbitrary mean and covariance matrix) over a triangle in the plane. I would like the solution to work over a wide range ...
0
votes
3answers
42 views

integrating $A^2=\frac{1}{2\pi}\int^\infty_{-\infty}\int^\infty_{-\infty}e^{-\frac{y^2+z^2}{2}}dydz$

When proving that $$\int^{\infty}_{-\infty}\frac{1}{\sqrt{2\pi}\sigma}{e^{-\frac{1}{2}({\frac{x-\mu}{\sigma})}^2}}dx=1$$ and I faced a problem, ...
3
votes
2answers
234 views

Characteristic Function of Inverse Gaussian Distribution

The pdf of Inverse Gaussian distribution, IG$(\mu,\lambda)$, is : $$p_X(x)=\sqrt\frac{\lambda}{2\pi x^3}\exp\left[\frac{-\lambda}{2\mu^2x}(x-\mu)^2\right];\quad x>0,\lambda,\mu>0$$ I have to ...
6
votes
3answers
1k 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 ...
2
votes
0answers
90 views

Can any one help me normalize this equation? (Modified 3D Gaussian)

$$\exp\left( - e^{d-sz} - 2 \left( \frac{z^2}{r^2f^2}+\frac{x^2+y^2}{r^2} \right) \right)$$ Note if this equation can't be normalized another equation with similar proprieties would also be ...
1
vote
2answers
220 views

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

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

How to integrate $\int_n^{+\infty} x \exp\{-ax^2+bx+c\}dx$?

How can I integrate, $$ \int_n^{+\infty} x \exp\{-ax^2+bx+c\}dx $$ and what's the result w.r.t the Gaussian function's p.d.f $p(x)$ and c.d.f $\phi(x)$? Thanks!
1
vote
2answers
311 views

Definite integral of cdf of the form $\Phi(\alpha+\sqrt{d^2-\frac{x^2}{2\sigma^2}})$

Any solution for the following definite integaral? Here $\Phi(x)$ represents the cumulative distributive function of standard normal distribution ...