For questions on bivariate distributions, the combined probability distribution of two randomly different variables.

Learn about bivariate distributions:

For questions on bivariate distributions, the combined probability distribution of two randomly different variables.

Learn about bivariate distributions:

272 questions

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$X$ 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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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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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

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Suppose $X$ and $Y$ are continuous random variables with joint p.d.f.
$$f(x,y) = e^{-y},\,\, 0

user153009

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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.

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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

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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

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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

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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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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

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$X_1$ is a sample from a normal distribution with mean$=\mu$ and variance $= 1$. The joint distribution of $X_1$ and the sample mean is bivariate normal. I need to find the conditional distribution of $X_1$ given the sample mean. To do this I need…

Tim

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I would like to generate points (x,y) in a 2-D plane that has a circular normal distribution similar to this:
I found multiple terms for describing a "circular normal distribution" and yet, I'm not sure which one to use (let me know if bivariate…

m_power

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Let $X$ and $Y$ be random variables with probability density $f(x, y)=12xy(1-y)$ if $0

Max

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I was asked the following question by my professor. Suppose $x$ and $y$ are jointly normal variables. We further suppose the marginal distribution of $x$ and $y$ are $N(0, 1)$ and $N(0, 2)$ respectively. Their correlation is $\rho$, what is…

Daniel

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Suppose that A and B are 2 r.v with joint probability distribution given by $f_{AB}(a,b) = -\frac{1}{2}(ln(a)+ln(b))$, if $0 \lt a \lt 1$ and $0 \lt b \lt 1$, and 0 otherwise.
Define $X = A+B$ and $Y = \frac{A}{B}$.
How do I derive the joint…

user986741