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Definition 8.1.1. Expectation. The expected value or expectation, E(x), of a random variable
x is its mean average value in the population. We denote the expectation of x by µ, such that
E(x) = µ.
Now that we have the definition of expectation we can define the variance, which characterises
the "spread" of a random variable:
Definition 8.1.2. Variance. The variance of a random variable is the expectation of the squared
deviations of the variable from the mean, denoted by σ 2 (x) = E[(x − µ) 2 ].
Notice that the variance is always non-negative. This allows us to define the standard deviation:
Definition 8.1.3. Standard Deviation. The standard deviation of a random variable x, σ(x), is
the square root of the variance of x.
Now that we’ve outlined these elementary statistical definitions we can generalise the variance
to the concept of covariance between two random variables. Covariance tells us how linearly
related these two variables are:
Definition 8.1.4. Covariance. The covariance of two random variables x and y, each having
respective expectations µ x and µ y , is given by σ(x, y) = E[(x − µ x )(y − µ y )].
Covariance tells us how two variables move together.
Since we are in a statistical situation we do not have access to the population means µ x and
µ y . Instead we must estimate the covariance from a sample. For this we use the respective
sample means ¯x and ȳ.
If we consider a set of n pairs of elements of random variables from x and y, given by (x i , y i ),
the sample covariance, Cov(x, y) (also sometimes denoted by q(x, y)) is given by:
Cov(x, y) = 1
n − 1
n∑
(x i − ¯x)(y i − ȳ) (8.1)
i=1
Note: You may be wondering why we divide by n − 1 in the denominator, rather than n. This
is a valid question! The reason we choose n − 1 is that it makes Cov(x, y) an unbiased estimator.
8.1.1 Example: Sample Covariance in R
This will be our first usage of the R statistical language in the book. We have previously discussed
the installation procedure, so you can refer back to the introductory chapter if you need to install
R. Assuming you have R installed you can open up the R terminal.
In the following commands we are going to simulate two vectors of length 100, each with a
linearly increasing sequence of integers with some normally distributed noise added. Thus we
are constructing linearly associated variables by design.
We will firstly construct a scatter plot and then calculate the sample covariance using the
cor function. In order to ensure you see exactly the same data as I do, we will set a random
seed of 1 and 2 respectively for each variable: