Let's say I choose a number between 1-4 and the probability of choosing each number is as such:

p(1)=0.35

p(2)=0.25

p(3)=0.15

p(4)=0.25

I do this 100 times. Let X be the sum of all the numbers I picked. What is the approximate PDF of X?

Also, what is the probability that X > 200?

I think I have to use the normal distribution for the last question, but I'm not entirely sure how to go about setting this up and solving this. Any help would be greatly appreciated! Thanks.