I am new to this forum as well as to actuary field. I have this question:

There are two random variables X_1 and X_2 following standard normal distributions:

X_1 ~ N(0,1)

X_2 ~ N(0,1)

If we now set

Y_1 = X_1

Y_2 = X_1+X_2

then we know that

Y_1 ~ N(0,1)

Y_2 ~ N(0,2)

The question is how to find the joint CDF Of Y = (Y_1, Y_2). Can we find the joint CDF in a closed form? If yes, how?

This is a problem arising in the field of risk claims.

Is there anyone who can guide me in resolving this issue?

Many thanks in advance

Kind regards

hpriye