What is the life expectancy a patient with lung cancer that has spread to the bones and brain and is currently being treated with chemotherapy?

Not good. 6 months at the most. Give or take. Get affairs in order, hospice is a good option if available. Sorry and good luck!
Months to years. The answer will vary based on other co-existing health problems, the type of lung cancer (small cell, adenocarcinoma, or squamous) and whether or not the tumor has mutations that can be treated with targeted therapy. In a large trial of patients with advanced lung cancer without mutations, 50% of patients who received chemotherapy lived longer than 9 months (j clin oncol. 2012;30(24):3002).