How do you update an existing date to a random date in a range?

In one of my tables I have datetime field in which the data in the table column is populated with something like "2016-01-07 01:33:00".

What I want to do is change ONLY the date to a random date within a range (ie: 2016-02-01 thru 2016-02-28) without changing the time. The end result might be "2016-02-13 01:33:00".

What mysql command string would accomplish this task?