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hello!
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!
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!