Hi Alistu,
I agree with you that addition of just a magnitude (as any deterministic value) will just shift the signal (as you can see from your picture). As you mentioned in your second comment, any noise DOES defined by its distribution function. However, as you know, there are different sources of noise in p-i-n photodiode for different noises: thermal, short and dark. All these sources have different nature that defines their distribution function.