Java weighted random chance centered around a number

I have a program in which I need to generate random numbers that determine various outputs(To explain the exact reason would be too long). In theory a high number (lets say 100,000) is a valid output for my program, but its most likely(but not entirely impossible) going to end up being useless output.

I'd like to generate random numbers that are weighted to be around a "normalized" number. For example I'd pick a number (10), and the majority of numbers that are randomly generated will be near 10. But there's a small chance the random number could any integer. I currently just use a range when generating the numbers, but this bothers me since numbers outside this range could potentially be valid and useful output.

Is there an easy way to do this without introducing to much overhead or having to map a percentage chances to individual integers?