Mathematics

Questions tagged [bivariate-distributions]

For questions on bivariate distribution, the joint probability distribution of two random variables.

Learn about bivariate distributions:

300 questions
16
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2 answers

$P(X>0,Y>0)$ for a bivariate normal distribution with correlation $\rho$

$X$ and $Y$ have a bivariate normal distribution with $\rho$ as covariance. $X$ and $Y$ are standard normal variables. I showed that $X$ and $Z= \dfrac{Y-\rho X}{\sqrt{1-\rho^2}}$ are independent standard normal variables. Using this I need to show…
user669083
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8
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2 answers

Prove $\lim_{c\to\infty} P(X-\varepsilonc,Y>c)=1$ for all $\varepsilon>0$ if $X,Y$ are i.i.d. Normal$(0,1)$?

Suppose random variables $X$ and $Y$ are i.i.d. Normal $(0,1)$ . Consider the following events, where $\varepsilon>0, c>0$ : $$\begin{align*} Q&=\{(x,y)\in\Bbb R^2: x>c, y>c\}\\ C&=\{(x,y)\in Q:y<x-\varepsilon\}\\ D&=\{(x,y)\in…
r.e.s.
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6
votes
1 answer

Variance of Z = max(X,Y) where X Y are jointly bivariate normal

I have a question about the bivariate normal r.v.'s Given $X, Y \sim \operatorname{Normal}(0,1)$ with correlation coefficient $\rho$. Let $Z=\max(X,Y)$. Show that $\operatorname E Z^2=1$. My attempt: I found that $\operatorname E…
Harry
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5
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E[XY] where X and Y are the **sign functions** of standard normal distributions

The following question is from the book: "150 Most Frequently Asked Questions on Quant Interviews" By Stefanica, Radoicic, and Wang. Let $X$ and $Y$ be standard normal variables with joint normal distribution with correlation $\rho$ . Find the…
Joe
  • 563
5
votes
1 answer

Integral of a bivariate Gaussian in the positive quadrant

I am looking for a reference (or a somewhat simple proof) for the following result, which for instance Mathematica spits out without too much effort. Here $a,b,c \in \mathbb{R}$ are constants satisfying $a, c < 0$ and $b^2 < 4 a c$.…
TMM
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5
votes
1 answer

Jacobian Transformation p.d.f

Suppose $X$ and $Y$ are continuous random variables with joint p.d.f. $$f(x,y) = e^{-y},\,\, 0<x<y <\infty$$ (a) Find the joint p.d.f. of $U=X+Y$ and $V=X$ . Be sure to specify the support of $(U,V)$ . (b) Find the marginal p.d.f. of $U$ and the…
user153009
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4
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Joint distribution of two brownian bridges

Consider $X_t=B_t-tB_1$ for $0\leq t\leq 1$ , where $B_t$ represents the standard Brownian motion. I derived that $E(X_t)=0$ and $\operatorname{Cov}(X_t,X_s)=\min(t,s)-st$ . Now, I am asked the joint distribution of $X_t$ and $X_s$ for some fixed…
david_rios
  • 245
4
votes
0 answers

Estimating two means of bivariate gaussian

Consider a bivariate gaussian distribution, with parameters $\mu_1$ and $\mu_2$ for the two unknown means, and $\sigma_1$ , $\sigma_2$ and $\rho$ for the known covariance matrix, \begin{align} \Sigma&=\left(\begin{array}{cc} \sigma_1^2 & \sigma_{12}…
Daniel S.
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4
votes
1 answer

Asymptotically uncorrelated sequence converges in distribution to normal distribution?

If we know that the two random sequences $\{X_n\}$ and $\{Y_n\}$ $$ X_n \stackrel{d}{\longrightarrow} N(0,\sigma_1^2)\\ Y_n \stackrel{d}{\longrightarrow} N(0,\sigma_2^2) $$ and $\mathrm{cov}(X_n,Y_n) \to 0$, can we conclude that the joint random…
fun with math
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3
votes
1 answer

How to calculate $P(X+Y<0 | X<0)$?

For $X \sim N(\mu_x,\sigma_x^2)$ and $Y\sim N(\mu_y,\sigma_y^2)$ which are independent, can I calculate $P(X+Y<0 | X<0)$ ? I found this question for independent standard r.v and this one for correlated standard r.v , however I could not seem to…
coffee-raid
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3
votes
2 answers

The root of the sum of two normally distributed variables

Given $X,Y \sim \mathcal{N}(0,1)$ and $Z=\sqrt{X^2 + Y^2}$ , find the PDF of $Z$ . I know from digging around that this will follow a Rayleigh distribution since the sum of two squared normally distributed variables follow an exponential…
J N
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3
votes
1 answer

Derivation of bivariate Gaussian copula density

The multivariate Gaussian copula density , derived here , is $$c(u_1,\ldots,u_n;\Sigma)=|\Sigma|^{-\frac{1}{2}}\exp\!\left(-\frac{1}{2}x^{\top}(\Sigma^{-1}-I)x\right)$$ where $\Sigma$ is the covariance matrix, and…
develarist
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3
votes
3 answers

Distribution of $Z=\frac{X}{X+Y}$ when $X,Y$ are i.i.d geometric random variables

If $X,Y$ are independent and have same geometric distribution with parameter $p$ , find the distribution of $Z=\frac{X}{X+Y}$ . The solution is…
Lookout
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3
votes
2 answers

Finding Joint PMF of bivariate disttibution

Problem: Let $W$ equal the weight of laundry soap in a 1-kilogram box that is distributed in Southeast Asia. Suppose $P(W<1)=0.02$ and $P(W>1.072)=0.08$ . Call a box of soap light, good, or heavy depending on whether $W<1$ , $1 \leq W \leq 1.072$ ,…
spruce
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3
votes
4 answers

Conditional joint probabiltiy of a given pair

The bivariate PDF of a random pair $(X, Y)$ is given by: $f_{X,Y}(x,y) = 2e^{-x}e^{-2y}$ , $x\ge0, y\ge0$ What is the probability $Y < 4$ given $X > 1$? I calculated the conditional probability as $f_{Y\mid X}(y) = \frac{f_{X,Y}(x,y)}{f_X(x)}$ From…
user3067059
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