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

An exercise question asks

Let X1,X2 be rvs having a common Normal distribution N(0,1) with Corr(X1,X2)=ρ. Calculate the coefficient of upper tail-dependence for all ρ[1,1].

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

My first thought was that they meant both X1 and X2 are univariate normal 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?

Answer

It means that two things are true.

First:

P(X1<t)=P(X2<t)

for all real numbers t (i.e., X1 and X2 have the same distribution, often the shorthand equidistributed is used to describe this condition).

Second:

P(X1<t)=1σ2πte(xμ)22σ2dx

for some fixed numbers μ and σ (i.e. the distribution of X1 (*) is a normal distribution).

This doesn't imply that (X1,X2) 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 X2 is also a normal distribution.

Attribution
Source : Link , Question Author : FoetDen , Answer Author : Xi'an

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