## Time Average

We are interested in the manipulation of average of a sequence of measurement. Let us define the time average or tag in short as the arithmetic mean of the measurement sequence up to time as (1)

The diagram of time-average is shown in the figure below Diagram of time-average

Example :

We have sequence , then , ## Properties

### Addition or Subtraction of Two Averages

Now we have two sequences and , where , then addition or subtraction of the two averages at the same length will produce (2)

Proof: QED

Addition or subtraction of two averages at the same length is equal to summation or subtraction of element of the sequences at the same position. Addition of two averages is commutative because the order of the average is not important, . However, subtraction of two averages is non-commutative .

Example :

We have sequence and , then  ### Multiplication of Averages

Multiplication of two averages is equal to summation of product permutation of the two sequences times the inverse product of their length, that is (3)

Proof: QED

The lengths of the two sequences are not necessarily the same. The double sum indicates product permutation, which is multiplication of each element of the first sequence to each element of the second sequence, both up to the specified length of the sequences.

Multiplication of average is commutative. The order of the average is not important. Example :

We have sequence and , then  If none of elements in sequence is allowed to be zero, then division of two averages is possible: (4)

Division of two averages is not commutative, .

Multiplication of three averages is easily extended from the two averages, that is (5)

If we define Permutation products as (6)

Then, we can generalize the multiplication of averages as (7)

### Distributive Law of Averages

Suppose we have averages of three sequences of the same length, then we can have distributive law of average as (8)