# Tagged Questions

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Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition $$T(x_1,\ldots,x_n) := ... 1answer 94 views ### Laplace transform of : t^{\gamma-1} F(\alpha,\beta,\delta,t), where F is the Gauss' hypergeometric function What is the Laplace transform of : t^{\gamma-1} F(\alpha,\beta,\delta,t), where \gamma >0  and F is the Gauss' hypergeometric function. Thanks! 2answers 169 views ### How to calculate P(\sum_{i=1}^{m}(A_i+S_i)\le L) with A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu) and positive integers \lambda\neq\mu? Recently I was stumped by the calculation of the probability$$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$where A_i \sim \text{exp}(\lambda), S_i \sim ... 1answer 153 views ### Is any derivative of f_1^x f_0^{1-x} w.r.t. x integrable? For f_0 and f_1 two continuos probability density functions on \mathbb{R}, by HÃ¶lder, I know that f_1^x f_0^{1-x} is integrable on \mathbb{R}, where 0 \leq x \leq 1. Let l=f_1/f_0, then ... 0answers 105 views ### Marginalizing multivariate normal over defined interval Hello everyone, I am trying to obtain an analytic expression for the following Gaussian integral$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} ...
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 ...
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 ...