Is there a similar concept to standard normal distribution in the context of joint probability?

Question

The simplest case of a normal distribution is known as the standard normal distribution

which is described by this probability density function:

${\displaystyle \varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}}$

this is for one random variable.

how about 2 random variables, namely, is there a similar concept for joint probability, something like standard normal joint probability distribution?

I've searched this on math.stackexchange.com and got '0 results'.

any clue?

Answer 1

This is called The Standard Bivariate Normal Distribution

a pair of independent random variables X and Y, each follows the standard normal distribution ${\displaystyle X \sim \ {\mathcal {N}}(0,1)}$, and ${\displaystyle Y \sim \ {\mathcal {N}}(0,1)}$

For a constant $\rho$ with $-1< \rho <1$, define random variables

and then

more info is here.