MHB Which is the distribution of (X,Y) ?

  • Thread starter Thread starter mathmari
  • Start date Start date
  • Tags Tags
    Distribution
AI Thread Summary
The random variables X and Y follow a standard normal distribution N(0,1) and are independent, resulting in a bivariate normal distribution for the pair (X,Y). The covariance table for (X,Y) indicates that the covariance is zero due to their independence. It is clarified that (X,Y) itself is not a linear combination of X and Y. However, any linear combination of the form λX + μY will still yield a normal distribution. Understanding these properties is essential for analyzing the relationship between independent normal variables.
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :o

We have that $X$ and $Y$ follow the normal distribution $N(0,1)$ and are independent.
  1. Which is the distribution of $(X,Y)$ ?
  2. Which is the covariance table of $(X,Y)$ ?

Is $(X,Y)$ related to the linear combination of $X$ and $Y$ ? (Wondering)
 
Physics news on Phys.org
mathmari said:
Hey! :o

We have that $X$ and $Y$ follow the normal distribution $N(0,1)$ and are independent.
  1. Which is the distribution of $(X,Y)$ ?
  2. Which is the covariance table of $(X,Y)$ ?

Is $(X,Y)$ related to the linear combination of $X$ and $Y$ ?

Hey mathmari! (Wave)

It's a so called Bivariate normal distribution.
The section in the wiki article explains how to get the covariance table. (Thinking)

And no, $(X,Y)$ is not linear combination of $X$ and $Y$.
However, a property of a bivariate normal distribution is that any linear combination $\lambda X + \mu Y$ has a normal distribution.
 
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
Back
Top