Tagged Questions

36 views

How to update the probabilities so that it still sum up to $1$?

At time $t$, I have a probability vector $\mathbf{\pi}^{t}=\left({\pi}_{1}^{t}, \cdots, {\pi}_{n}^{t} \right)$. I would like to construct a function $f(\cdot)$ and update the vector ...
69 views

Copulas and their properties

I am working with the following copula, and have a few questions about it: $C(x,y) = xy + \theta (1-x)(1-y)xy$ Here $\theta \in [-1,1]$ and $x,y \in [0,1]$ First, I am trying to show this copula is ...
147 views

Mean and variance of truncated generalized Beta distribution

The generalized Beta probability density function is given by: $$f(x) = \frac{(x-A)^{\alpha - 1} (B-x)^{\beta - 1}}{(B-A)^{\alpha + \beta - 1} \mathrm{B}(\alpha ,\beta)}$$ for $A<x<B$, and ...
91 views

Copulas, implication

Let $C$ be a copula function. Prove that $C(t,1-t)=0$ for all $t\in[0,1]$ implies that $C(u,v)=\max(u+v-1,0)$. I think the implication other way around is easy to see, however I can't see why the ...
60 views

Autocorrelation functions of 2 correlated stationairy processes

I have some trouble solving the following problem: Given are the stationairy processes $X_t$ and $Y_t$: $X_t = Z_t*\sqrt{7+0.5X_{t-1}^2}$ $Y_t = 2+(2/3)*Y_{t-1}+X_t$ Where $Z_t$ is distributed IID ...
74 views

Fourier transform of displaced airy function

I need to find the fourier transform of displaced airy function.The function is $ψ_n(ξ) = N_n \text{Ai}(ξ − ξ_n)$, where $ξ=x/x_0$, $x_0=(1/2)^{1/3}$, $ξ_n = (3\pi/2)(n − 1/4)^{2/3}$ and $N_n$ is ...
115 views

4k views

Why is the error function defined as it is?

$\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of ...
224 views

expectation of incomplete gamma

Is the expectation of the (upper/lower) incomplete gamma function known? $$\int_0^{+\infty} x \Gamma(A, x) \mathrm dx$$
Given $\frac{a}{x-1} \leq \frac{b}{y-1} \leq \frac{c}{z-1}$ with $a,b,c > 0$ and $x,y,z > 1$, I want to show that \frac{(\frac{a}{a+b})^{x-1}(\frac{b}{a+b})^{y-1}}{B(x,y)\cdot (x+y-1)} + ...