For questions on bivariate distribution, the joint probability distribution of two random variables.
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Questions tagged [bivariate-distributions]
300 questions
15
votes
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
- 1,189
8
votes
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 $\varepsilon>0, c>0$:
$$\begin{align*}
Q&=\{(x,y)\in\Bbb R^2: x>c, y>c\}\\
C&=\{(x,y)\in Q:y
r.e.s.
- 15,537
6
votes
1 answer
Variance of Z = max(X,Y) where X Y are jointly bivariate normal
I have a question about the bivariate normal r.v.'s
Given $X, Y \sim \operatorname{Normal}(0,1)$ with correlation coefficient $\rho$. Let $Z=\max(X,Y)$. Show that $\operatorname E Z^2=1$.
My attempt:
I found that $\operatorname E…
Harry
- 631
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- 16
5
votes
0 answers
E[XY] where X and Y are the **sign functions** of standard normal distributions
The following question is from the book: "150 Most Frequently Asked Questions on Quant Interviews" By Stefanica, Radoicic, and Wang.
Let $X$ and $Y$ be standard normal variables with joint normal distribution with correlation $\rho$.
Find the…
Joe
- 563
5
votes
1 answer
Integral of a bivariate Gaussian in the positive quadrant
I am looking for a reference (or a somewhat simple proof) for the following result, which for instance Mathematica spits out without too much effort. Here $a,b,c \in \mathbb{R}$ are constants satisfying $a, c < 0$ and $b^2 < 4 a…
TMM
- 10,065
5
votes
1 answer
Jacobian Transformation p.d.f
Suppose $X$ and $Y$ are continuous random variables with joint p.d.f.
$$f(x,y) = e^{-y},\,\, 0
user153009
- 642
- 1
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- 16
4
votes
0 answers
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 &…
Daniel S.
- 863
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…
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
- 31
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
- 1,584
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…
Lookout
- 2,221
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- 31
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 $W<1$, $1 \leq W \leq 1.072$, or…
spruce
- 695
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
- 289
3
votes
2 answers
How does one solve a bivariate normal density function?
If the exponent of $e$ of a bivariate normal density is
$$\frac{-1}{54} *(x^2+4y^2+2xy+2x+8y+4) \\\text{find } \sigma_{1},\sigma_{2} \text{ and } p \text{ given that } \mu_{1} =0 \text{ and } \mu_{2}=-1. $$
One must use this definition to…
Jon
- 1,916