1
vote
1answer
35 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 ...
0
votes
1answer
57 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 ...
0
votes
1answer
86 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 ...
1
vote
2answers
86 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 ...
0
votes
1answer
57 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 ...
1
vote
0answers
66 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 ...
1
vote
0answers
108 views

Limiting behavior of an integral involving incomplete Gamma function

I am wondering about the limiting behavior as $k\rightarrow\infty$ of the following integral: $$I(k)=\frac{2^{-k/2}}{\Gamma(k/2)}\int_{f(k)}^\infty ...
5
votes
0answers
206 views

Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: \begin{equation} I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt, \end{equation} \begin{equation} ...
1
vote
1answer
33 views

Cardinality of Solutions to an Inequality [duplicate]

Show that the number of solutions in nonneg. int. of the ineq. $$x_1+x_2+\cdots +x_n\leq M,$$ where $M$ is a nonneg. int., is $C(M+n,n)$.
1
vote
1answer
37 views

Logarithmic Power Series?

Take $\lambda>0$ and $0<x<1$. What sort of insight does anybody have on the function $f(\lambda,x)=\lambda\sum_{k=0}^{\infty} \sum_{j=0}^k ...
0
votes
1answer
174 views

Expected value of a Poisson sum of confluent hypergeometric functions

How to compute the expected value of a Poisson sum of the following confluent hypergeometric function: $$ \sum_{y=1}^{Y} {}_1F_1(y,1,z) $$ where y is positive integer taking values from the Poisson ...
1
vote
0answers
89 views

Inequalities involving regularized incomplete Gamma functions

I am new to the world of the Gamma functions and am wondering if there exist positive functions $f_1(x)>0$ and $g_1(x)>0$, and non-negative functions $f_2(x)\geq0$ and $g_2(x)\geq0$ such that ...
1
vote
0answers
45 views

The cdf of a beta variable, evaluated at the mean

Consider a Beta random variable $X$ with shape parameters $k/2$ and $(d-k)/2$, where the parameters $k, d$ are integers that satisfy $0 < k < d$. What is the best possible upper bound for the ...
5
votes
1answer
149 views

Are these numbers $h_{r,s}$ irrational?

I came across these numbers in my work some time ago. This type of expressions do not exist in closed form (not to confuse with Vandermonde convolution), I already know that. To simplify I denote ...
2
votes
0answers
246 views

Determining the probability density function from an equation

I have the following (for me quite interesting) densities for which I am completely stuck. I only hope that you can provide me some help. Let me introduce my problem. I have two probability ...
1
vote
2answers
1k views

Expected value of $\ln X$ if $X$ is $\Gamma(a,b)$ distributed.

I'm new here and hope you can help. It's really late here in South Africa, maybe my mind just doesn't want to function now! But I need to figure out how to get a closed form expression hopefully for ...
1
vote
1answer
158 views

A uniqueness proposition involving Erf, the error function

This is a MathOverflow cross-post (currently no answer there) and a generalization of a previous MathOverflow question, "Reducing system of equations involving Erf, Error Function". Consider the ...
7
votes
2answers
718 views

Are there well known lower bounds for the upper incomplete gamma function?

Let $a >0, b >0$, and $r \in \mathbb{R}$. I am trying to find a lower bound for the integral $$\int_a^\infty y^{-r} \exp\left( - b(y-a)^2\right) \,\mathrm dy.$$ After consulting the Wikipedia ...
0
votes
1answer
145 views

Integration about standard normal

Let $N(x)$ denote the cdf of standard normal and $n(x)$ denote the pdf of standard normal. How to evaluate the integral $\int\limits_{-\infty}^\infty N(a+x) n(x) \mathrm{d} x$ ? Thanks a lot!
6
votes
1answer
208 views

Laguerre polynomials and inclusion-exclusion

Does anyone know a reference for the solution of the generalized derangement problem via Laguerre polynomials? The Wikipedia article here says that this is an application of inclusion-exclusion, but ...
23
votes
7answers
2k views

Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$

I would like your help with proving that for every $0 \leq k \leq n$, $$\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx . $$ I tried to integration by parts and to get a pattern or to ...
2
votes
0answers
160 views

Help with an integral inequality involving an incomplete beta function

I would like to determine if the following inequality is true: ...
9
votes
5answers
896 views

Proof that $x \Phi(x) + \Phi'(x) \geq 0$ $\forall x$, where $\Phi$ is the normal CDF

As title. Can anyone supply a simple proof that $$x \Phi(x) + \Phi'(x) \geq 0 \quad \forall x\in\mathbb{R}$$ where $\Phi$ is the standard normal CDF, i.e. $$\Phi(x) = \int_{-\infty}^x ...
29
votes
2answers
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 ...
1
vote
2answers
218 views

expectation of incomplete gamma

Is the expectation of the (upper/lower) incomplete gamma function known? $$\int_0^{+\infty} x \Gamma(A, x) \mathrm dx$$
5
votes
0answers
147 views

Bounding function involving Beta functions

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)} + ...
2
votes
0answers
46 views

Deriving functions for empiracal distributions -very applied mathatics

First I am a new user of this site. Second my math background is very limited, although I do have a lot of experience in applied statistics. Component or piece part failures on high value parts($1000 ...