# What does it mean to say that X1,X2X_1, X_2 have a “common” Normal distribution?

An exercise question asks

Let $$X1,X2X_1, X_2$$ be rvs having a common Normal distribution $$N(0,1)N(0,1)$$ with $$Corr(X1,X2)=ρ\operatorname{Corr}(X_1, X_2) = \rho$$. Calculate the coefficient of upper tail-dependence for all $$ρ∈[−1,1]\rho \in [-1, 1]$$.

What does it mean with it says they have a “common” Normal distribution?

My first thought was that they meant both $$X1X_1$$ and $$X2X_2$$ are univariate normal $$N(0,1)N(0,1)$$ distributed variables. However, if that is true, then the question doesn’t make sense. The tail-dependence cannot be calculated.

So I am left to believe that by “common” Normal distribution, they mean the bivariate Normal distribution?

It means that two things are true.

First:

$$P(X1

for all real numbers $$tt$$ (i.e., $$X1X_1$$ and $$X2X_2$$ have the same distribution, often the shorthand equidistributed is used to describe this condition).

Second:

$$P(X1

for some fixed numbers $$μ\mu$$ and $$σ\sigma$$ (i.e. the distribution of $$X1X_1$$ (*) is a normal distribution).

This doesn't imply that $$(X1,X2)(X_1, X_2)$$ is joint normal without further assumptions. If that was intended, it's not what the author actually wrote.

(*) Given the first condition, this implies that the distribution of $$X2X_2$$ is also a normal distribution.