Let X_1,...,X_n be non-negative independent random variables with mean u, and mean absolute deviation (MAD) d. Preferably I would like to say they are from the same ambiguity set and not iid, but if it is easier iid is alright. Define Y = X_1+...+X_n.
Clearly, u_Y = n*u, but d_Y is not so easy to compute. However, we know d_Y <= n*d from the triangle inequality, and I do know this bound is tight in this setting.
Can anybody provide for me a distribution for X1,...,X_n that attains this bound, if anybody happens to know? I would very much appriciate it.
Clearly, u_Y = n*u, but d_Y is not so easy to compute. However, we know d_Y <= n*d from the triangle inequality, and I do know this bound is tight in this setting.
Can anybody provide for me a distribution for X1,...,X_n that attains this bound, if anybody happens to know? I would very much appriciate it.