You might want to try these on your own as they are fairly standard. But -
Since the two bulbs are independent, the pdf of the two is just the product of the individual pdfs (which are the same save for the variable name). However you are interested in Y=X1+X2. I would use the MGF approach here (hint, it's a Gamma), and then simply refer to the cdf of a Gamma.
If you do not know that then the change of variables - Y=X1+X2, X2=X2 - may be the easier route.