How long to find out if a death looks suspicious?

A family member died last weekend. He was allegedly found dead, lying down his porch steps as if he'd fallen. His body had been robbed and money, medication and other items were missing from inside his home, and yet the police did not suspect that he was murdered, but that he had died of a stroke or heart attack and was robbed after the fact. Two people were soon arrested for larceny, and today the family received a coroner's report saying that he had died from asphyxiation by constriction. Did the police handle this properly? Shouldn't there have been an investigation immediately? I don't even know what I'm trying to ask here. I'm so upset. I guess, what would the proper procedure be in such a case?