00:01
Here we are going to show that if a discrete random variable takes values on the positive integers, then its expectation can be expressed as the following.
00:11
Let's start from the definition of expectation for a discrete random variable.
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Some over all possible values it can take, which is all the positive integers.
00:23
Now let's write out this expansion very explicitly.
00:27
The first term is 1 times the probability of x equals 1.
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The second term is 2 times the probability of x equals 2, which we can write as the probability of x equals 2 plus the probability of the x equals 2.
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The third term is 3 times the probability of x equals 3, which is, and so on.
01:05
Now, this expansion corresponds to doing the sum for each row first.
01:14
We can instead consider first summing over columns, so we can express this expectation as the sum of the fifth column, and then the sum of this column, plus the sum of this column, and so on.
01:34
Now, what is the first column? it's the probability for x to be 1, 2, 3, 4 up to infinity, and that's simply the probability of x being greater or equal than 1...