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Roger V.
Roger V.
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integration Integration of product of a gaussian pdf and a student-t pdf

I want to perform the following integration wrt $x$: $$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$$$\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(y-hx)^2}{2\sigma^2}}\left(1+\frac{x^2}{b}\right)^{-(\frac{b+1}{2})}dx$$

Here first part is a gaussian random variable (treating $y,h,\sigma$ as constants and the second part is student-t distribution with parameter $b$. Does this have any closed form or represents any other kind of distribution. I am unable to figure out.

integration of product of a gaussian pdf and a student-t pdf

I want to perform the following integration wrt $x$: $$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$

Here first part is a gaussian random variable (treating $y,h,\sigma$ as constants and the second part is student-t distribution with parameter $b$. Does this have any closed form or represents any other kind of distribution. I am unable to figure out.

Integration of product of a gaussian pdf and a student-t pdf

I want to perform the following integration wrt $x$: $$\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(y-hx)^2}{2\sigma^2}}\left(1+\frac{x^2}{b}\right)^{-(\frac{b+1}{2})}dx$$

Here first part is a gaussian random variable (treating $y,h,\sigma$ as constants and the second part is student-t distribution with parameter $b$. Does this have any closed form or represents any other kind of distribution. I am unable to figure out.

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Pikaboo
Pikaboo
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I want to perform the following integration wrt $x$: $$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$

Here first part is a gaussian random variable (treating $y,h,\sigma$ as constants and the second part is student-t distribution with parameter $b$. Does this have any closed form? or represents any other kind of distribution. I am unable to figure out.

I want to perform the following integration wrt $x$: $$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$

Here first part is a gaussian random variable (treating $y,h,\sigma$ as constants and the second part is student-t distribution with parameter $b$. Does this have any closed form?

I want to perform the following integration wrt $x$: $$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$

Here first part is a gaussian random variable (treating $y,h,\sigma$ as constants and the second part is student-t distribution with parameter $b$. Does this have any closed form or represents any other kind of distribution. I am unable to figure out.

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Pikaboo
Pikaboo
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  • 1

integration of product of a gaussian pdf and a student-t pdf

I want to perform the following integration wrt $x$: $$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$

Here first part is a gaussian random variable (treating $y,h,\sigma$ as constants and the second part is student-t distribution with parameter $b$. Does this have any closed form?