Yes, this is possible - you should be prepared however to independently find out about the related requirements and practicalities concerning your family members' residence in concerned Schengen country. For example, the family member(s) moving to Schengen with you must have sufficient means of support in order to be issued with a residence permit. You must check all the permit-related requirements with the immigration authorities, either the concerned Schengen country Embassy in your home country or the concerned Schengen country Immigration Service Migration
Also you should check with your hosting Schengen university/UAS about possible family student housing options. And if you bring school-aged children with you, you should contact the school education providers at your study location in Finland for information on your children's schooling options - similarly for possible daycare services for younger children.