0
votes
2answers
22 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
0
votes
1answer
28 views

Find marginal and conditional distributions [closed]

Consider the probabiility density function $f_{X_1, X_2}(x_1, x_2) = \left\{\begin{matrix}\frac{1}{8x_2} \exp\left\{ -\left( \frac{x_1}{2x_2} + \frac{x_2}{4}\right)\right\}, & x_1 > 0, ...
2
votes
0answers
35 views

General formula for dependent probability distributions

Recently I encountered the following problem: What is the mean distance between two random points on a unit square? I understand pen and paper methods for solving this exist however I'm ...
5
votes
1answer
49 views

Probability roots of quadratic lie in unit disc

$A,B\sim\mathscr{U}(0,1)$ and independent. We consider: $$x^2+2Ax+B=0$$ Given that both of the roots of this equation are real, what is the probability that they lie in the unit disc? ...
0
votes
1answer
27 views

systematic way of finding the bounds for change of variables (multivariable case), Jacobian

Let's say that $X,Y$ are independent standard normal random variables. I am interested in the distribution $P(X+Y\le 2t)$. Clearly, the domain of integration in this case is $-\infty<x<\infty$ ...
0
votes
0answers
29 views

multiplying Gaussian distributions of different dimensions

The multiplication of multivariate Gaussian distributions defined over some parameter vector of a given dimension can be achieved by the following. Assuming that the Gaussian is parametrized by the ...
3
votes
0answers
68 views

Understanding the setup for the probability that $Ax^2+Bx+C$ has real roots if A, B, and C are random variables uniformly distribted over (0,1).

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $Ax^2 + Bx + C$ has real roots? First, I set $P(B^2 - 4AC ...
2
votes
0answers
91 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
2
votes
0answers
55 views

Difficult multivarate random variable - how to calculate it?

I have a random variable defined by $Y=\frac{\sum_{j=1}^{N}l_j \cos\theta_j}{\sum_{j=1}^{N}l_j\sin\theta_j}$ where $l_j \sim \text{log-normal-distribution} (\left \langle l \right \rangle, \sigma _l)$ ...
0
votes
1answer
58 views

How to obtain a pdf of a random variable defined as a function of many variables?

Given $N$ independent random variables ($X_1$,$X_2$,...,$X_N$) with individual pdfs $f_1$,...,$f_N$: How to determine the pdf of a random variable $Y=G(X_1,...,X_N)$?
1
vote
4answers
105 views

Marginal density function understanding

Given a plane with three points, $(0, -1)$, $(2,0)$, and $(0, 1)$ with $x$-axis and $y$-axis connecting three points to make a triangle. Suppose this triangle represents the support for a joint ...
1
vote
0answers
101 views

Simplify the expectation of the maximum of two random variables

My aim is to simplify the maximum of two expressions each of which are a function of exponentially distributed random variables Given: positive constants $a,b,c,d$. Independent random variables $x,y ...
1
vote
0answers
98 views

Average arc length between two random points on a unit sphere?

I'm trying to find the average arc length between two random points on a unit sphere. The solution I've come up with is rather ugly. Consider a parametric surface: $$X(u,v)=\sin u\cos v\\Y(u,v)=\cos ...
0
votes
1answer
69 views

Creating a function from known data and variable relationships

I'm developing a game and I need to create a predictable function while most of the variables are not 100% under my control. I will explain the practical situation: You have two characters, trying to ...
1
vote
1answer
34 views

Kronecker delta for multivariate distributions?

I have found a formula (Theorem 2.1 here: http://arxiv.org/pdf/0905.4131v1.pdf) which shows the covariance matrix of a multivariate normal distribution $\Sigma_P$, but I'd like some help interpreting ...
1
vote
1answer
74 views

A modified Buffon's needle

A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it. What is the probability that the needle lies between the two lines? I can see how ...
2
votes
2answers
158 views

Other way to express $e^{|x|+|y|}$

I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...
3
votes
0answers
99 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
1answer
59 views

Marginal Probability Density: Integrand Values

I have a joint probability density function, $f(x,y)$. However, I have a constraint associated: $0< x < y < +\infty$. So, when I calculate the marginal probability densities, how do I ...
1
vote
1answer
138 views

$e^{F(x,y)}$ Type Multi-variable Exponential Integrals

I am sure all you integration buffs can do this faster than I can type it. Your help with a quick explanation and solution is appreciated. $$F _{XY} = \int_0^\infty\int_0^\infty xye^{-\frac{x^2 + ...
2
votes
1answer
401 views

What is the analytic expression for PDF of joint distribution of two Gaussian random vectors?

I know that if $X$ and $Y$ are random variables with respective PDFs, $$ f_X(x) = \frac{1}{\sqrt{2\pi\sigma_x^2}}\exp\left\{-\frac{\left(x-\mu_x\right)^2}{2\sigma_x^2}\right\} \\ f_Y(y) = ...
6
votes
1answer
139 views

Characterizing a distribution by its projections

Consider the density $f(x,y)=\large\frac{1}{2\pi}\frac{1}{\sqrt{1-x^2-y^2}}$ on the unit disk centered at the origin. There is a particular characterization of this distribution: it is the unique ...
0
votes
1answer
228 views

Help solving CDF for transformation of $ \ge 2 $ random variables or if it's impossible.

Suppose independent random variables $U, T$. Let $U$ have continuous Uniform distribution over $(0, 2\pi)$. Let $T$ have Exponential distribution with $\lambda = 1$. Let random variable $Y$ be a ...
6
votes
1answer
1k views

Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
0
votes
1answer
241 views

Degrees of freedom in a Multivariate t Distribution?

Can I have different degrees of freedom for each dimension in a Multivariate t Distribution? The functions that I use in Matlab (mvtpdf) and Mathematica (MultivariateTDistribution) accept only one ...
2
votes
1answer
644 views

Iterated Integrals and Unbounded Regions

Context I am having difficulty finding the posterior distribution of a Bayesian model with two parameters, which involves evaluating a double integral over an unbounded region. I prefer not to post ...
1
vote
2answers
393 views

Partial Derivative of Poisson CDF

I am trying to figure out how to take the partial derivative of the Poisson CDF $F(k, \lambda)$ with respect to $k$. I have seen the partial derivative taken with respect to $\lambda$, but am unsure ...
4
votes
1answer
2k views

Multivariate Normal Difference Distribution

Since the distribution of a difference of two normally distributed variates X and Y with means and variances $(\mu_x,\sigma_x^2)$ and $(\mu_y,\sigma_y^2)$ respectively is given by another normal ...