The most important thing to consider is when the income on the investments was earned.
If it was earned or accrued on the date of death or prior then it is included on the decedent's final tax return.
However, if the investment income was earned or accrued after the date of death then it is included in the estate of the decedent.
Our article, Death in the Family, has additional helpful information. My condolences for your loss.