Write formula for mean using only dot product: Σ xiyi=<vector(x),vector(y)>. Hint use vector(1)
To derive the formula for the mean using only the dot product and the concept of vectors, we can start by representing the set of values xi and yi as vectors x and y, respectively. We can construct these vectors by concatenating the values with a vector of ones.
Let's say we have a set of n values:
x = [x1, x2, x3, ..., xn] and
y = [y1, y2, y3, ..., yn].
We can represent these vectors in terms of their dot product using the following formula:
Σ(xiyi) = <x, y> = Σ(xi * yi)
where Σ denotes the sum and < , > denotes the dot product.
To compute the mean, we divide this sum by the total number of values, n.
Mean = Σ(xiyi) / n = <x, y> / n
Since vector x is composed of the values xi and a vector of ones, it can be expressed as:
x = [x1, x2, x3, ..., xn, 1] = [xi, 1]
Similarly, vector y can be expressed as:
y = [y1, y2, y3, ..., yn, 1] = [yi, 1]
Substituting these values into the previous formula, we get:
Mean = <x, y> / n = Σ(xi * yi) / n
This is the formula for the mean using only the dot product.