Big sum and big product

In statistics, we use the big sum \(\sum\) and big product \(\prod\) notation a lot to compactly express the sum and product over multiple variables/parameters. For this reason, it might be worth to go over what exactly these things mean.

Say we are summing three numbers

\[x_1+x_2+x_3\]

Another way of writing this more compactly is:

\[\sum_{i=1}^3 x_i\]

This notation implies that the subscript \(i\) takes on the value of 1, 2, and 3. While you may not see much of an advantage of doing this in this example, imagine if we were summing 100 numbers. This would be simple using this latter notation:

\[\sum_{i=1}^{100} x_i\]

Here, our expression implies that the subscript \(i\) takes on the value of 1, 2, …, 100. As you can imagine, explicitly writing each element of the sum would be daunting.

The big product sign is the equivalent of the big sum sign but for multiplication. Say we are multiplying three numbers:

\[y_1 \times y_2 \times y_3\]

This can be written as:

\[\prod_{i=1}^3 y_i\]

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