0
votes
0answers
15 views

Intuition about the where the beta distribution has its maximum

I've stumped myself trying to develop an intuition about why the beta distribution should have its maximum where it does. I can differentiate easily enough, and I can manage a simple argument based on ...
4
votes
2answers
97 views

Does anyone know how to calculate the following integral?

Consider the function (coming from a joint probability density): $$ f(x,y) = \frac{1}{y}e^{-y-\frac{x}{y}}. $$ I want to evaluate the definite integral (marginal): $$ F(x) = \int_0^\infty f(x,y)\,dy. ...
2
votes
1answer
144 views

Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $

Please, help me with studying of useful practical features of the following functional space: $$\int_{-\infty}^\infty f(x) \, dx = 1$$ For example: 1) What basis types are most convenient for ...
2
votes
1answer
111 views

Proving a function (involving Erf) has a unique real root

I am trying to prove that the following function: $$g(x) = x - \sqrt{\frac{2}{\pi}} e^{-x^2/2} ...
1
vote
0answers
74 views

Fabius function and equivalent

The Fabius function $F$ can be defined on $[0,1]$ by $F(0)=0$ $F(1)=1$ on $[0,\frac{1}{2}]$ $F'(x)=2.F(2x)$ on $[\frac{1}{2},1]$ $F'(x)=2.F(2(1-x))$ It's a known example of a not analytic ...
3
votes
0answers
102 views

Useful approximation of the pdf

Good day to everyone. In my research work I came out with a function, which looks like this (it is the pdf of some random variable): $$f(x,\rho,\psi)=\frac{2}{\pi }+\sqrt{\frac{2}{\pi }} ...
2
votes
0answers
248 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 ...
7
votes
1answer
579 views

median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
1
vote
1answer
265 views

Help interpreting a gamma distribution

The following is from an article I'm reading and is the conditional density of a random variable that is distributed according to a gamma distribution, conditional on the value of a parameter $t$. ...
6
votes
2answers
288 views

Beta Function — finding a lower bound based on parameters

I would like to show that $$ 1-\frac{1}{c}Beta\left(c+1,\frac{1}{c}\right) \geq \frac{1}{c+1}.$$ for all $c \geq 2$. I have plotted it out for $c$ up through 200, and it seems to hold. Does anyone ...
1
vote
0answers
80 views

Computing the PDF of a product of the sum of 2 Nakagmi-m R.V.s with a Normal R.V

I really have two questions: One is about computing a PDF and the second is about how to sum a series involving $K_v(x)$ that the PDF in question seems to contain. I have come across the following ...
3
votes
1answer
232 views

Conceptually, what is the difference between the Beta function and the Beta distribution?

I have read the Wikipedia pages on the Beta function and the Beta distribution, but I'm still not sure I have a good intuition for what's going on. I'm am hoping someone will be kind enough to ...