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I'm trying to understand the following property:

Let X and Y be independent random variables z: = X + Y. Then

where fZ (z) is the probability mass function for a discrete random variable defined as follows:

and Wx is the set of possible values for the random variable X

Proof: Using the Law of total probability, which is this:

we obtain

Question-I don't see how Law of total probability helps us. I even don't understand how we get the first line of the proof-End of Question

Then my book proposes this example: We roll two dices. Let X/Y random variables, which indicate the number of points in the first/second die. We calculate the density of Z: = X + Y:

For 2 <= z <= 7 we obtain

And 7 <z <= 12:

Here I just get that:

But I do not understand why we have this index of summation nor why we put min and max in the next step

Could you help me please? Thank you!