Product Notation

When we want to multiply a long list of numbers, writing them all out becomes tedious. Product notation lets us write such products compactly using the Greek capital letter pi, $\prod.$

We write

$$\prod_{i=1}^{n} a_i = a_1 \cdot a_2 \cdot a_3 \cdots a_n.$$

Here $i$ is the index of multiplication, $1$ is the lower limit, $n$ is the upper limit, and $a_i$ is the factor, which depends on $i.$

For example,

$$\prod_{i=1}^{4} i = 1 \cdot 2 \cdot 3 \cdot 4 = 24.$$

The lower limit need not be $1$ — we may start the product at any integer. For instance,

$$\prod_{k=3}^{5} k^2 = 3^2 \cdot 4^2 \cdot 5^2 = 9 \cdot 16 \cdot 25 = 3600.$$

Product notation is the multiplicative analogue of summation notation $\sum.$ It appears throughout probability and statistics, especially when expressing the joint probability of several independent observations as a single product.