Fred is a U$.$S$.$ citizen. His wife, Christina, is a permanent resident of the US with a green card. Fred died in $2019.$ At the time of his death, he owned a life insurance policy on his life. It had a death benefit of $$500,000$ and cash value of $$200,000.$ Christina is the beneficiary.

a$)$ Will any amount of the life insurance be included in Fred's gross estate?

b$)$ If yes, what amount?

c$)$ Will any amount of the life insurance qualify for the unlimited marital deduction?

d$)$ If yes, what amount?