0
votes
1answer
96 views

Is any derivative of $f_1^x f_0^{1-x}$ w.r.t. $x$ integrable?

For $f_0$ and $f_1$ two continuos probability density functions on $\mathbb{R}$, by Hölder, I know that $f_1^x f_0^{1-x}$ is integrable on $\mathbb{R}$, where $0 \leq x \leq 1$. Let $l=f_1/f_0$, then ...
2
votes
0answers
93 views

Marginalizing multivariate normal over defined interval

Hello everyone, I am trying to obtain an analytic expression for the following Gaussian integral $$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} ...
1
vote
1answer
329 views

computing an integral involving standard normal pdf and cdf

recently, i need to compute this kind of integral: $$ \int ^\infty _c \Phi(ax+b) \phi(x) dx$$ where a, b and c are all constants and $\Phi(x)$ denotes the CDF of standard normal distribution and ...
2
votes
3answers
3k views

Integration of the product of pdf & cdf of normal distribution [closed]

Denote the pdf of normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int \phi(x) \Phi(\frac{x -b}{a}) dx$? Notice that when $a = 1$ and $b = 0$ the answer is ...