Reduce oven temp to 300F.
Using an electric mixer with the paddle attached, mix the cream cheese, sugar, flour and pudding mix with an electric mixer until combined.
Add eggs one at a time, beating slowly and scraping the sides of the bowl after each.
Add the mashed banana and vanilla. Mix on low speed until well combined, but still lumpy.
Pour the filling into the cooled crust.
Wrap outside of pan in a double layer of foil, covering the underside and extending all the way to the top.
Set wrapped pan in a large roasting pan, and pour hot water into roasting pan—to a depth of 2 inches or about halfway up the sides of cheesecake pan.
Bake for 90 minutes. Turn off heat and leave in oven with door closed for 20 minutes.
Crack open the oven and leave the cheesecake in the oven for another 20 minutes or until set.
Remove from oven and chill overnight.