Variance and Standard Deviation of a Random Variable

We have already looked at Variance and Standard deviation as measures of
dispersion under the section on
Averages. We can also measure the dispersion of Random variables across
a given distribution using Variance and Standard deviation. This allows us to better
understand whatever the distribution represents.

The Variance of a random variable X is also denoted by σ;²
but when sometimes can be written as Var(X).

Variance of a random variable can be defined as the expected value of the square
of the difference between the random variable and the mean.

Given that the random variable X has a mean of μ, then the variance
is expressed as:

In the previous section on
Expected value of a random variable, we saw that the method/formula for
calculating the expected value varied depending on whether the random variable was
discrete or continuous. As a consequence, we have two different methods for calculating
the variance of a random variable depending on whether the random variable is discrete
or continuous.

• For a Discrete random variable, the variance σ² is
  calculated as:

  

• For a Continuous random variable, the variance σ²
  is calculated as:

  

In both cases f(x) is the probability density function.

The Standard Deviation σ in both cases can be found by taking
the square root of the variance.

Example 1

A software engineering company tested a new product of theirs and found that the
number of errors per 100 CDs of the new software had the following probability distribution:

Find the Variance of X

Solution

The probability distribution given is discrete and so we can find the variance from
the following:

We need to find the mean μ first:

Then we find the variance:

Example 2

Find the Standard Deviation of a random variable X whose probability density function
is given by f(x) where:

Solution

Since the random variable X is continuous, we use the following formula to calculate
the variance:

First we find the mean μ

Then we find the variance as:

Simplifying the Variance formula

We have seen that variance of a random variable is given by:

We can attempt to simplify this formula by expanding the quadratic in the formula
above as follows:

We shall see in the next section that the expected value of a linear combination
behaves as follows:

Substituting the expanded form into the variance equation:

Remember that after you’ve calculated the mean μ, the result is a constant
and the expected value of a constant is that same constant.

This simplifies the formula as shown below:

but

which means that;

The above is a simplified formula for calculating the variance.

We can also derive the above for a discrete random variable as follows:

but since the total probability is 1

and

Therefore,

where by;

Hence

For a continuous random variable:

whereby

which means that

Variance of an Arbitrary function of a random variable g(X)

Consider an arbitrary function g(X), we saw that the expected value of this function
is given by:

• For a discrete case

  

• For a continuous case

  

The variance of this functiong(X) is denoted as σ_g(X)
and can be found as follows:

• For X is a discrete random variable

  

• For X is a continuous random variable

  

Covariance

In the section on
probability distributions, we saw that at times we might have to deal with
more than one random variable at a time, hence the need to study Joint Probability
Distributions.

Just as we can find the Expected value of a joint pair of random variables X
and Y, we can also find the variance and this is what we refer to as the
Covariance.

The Covariance of a joint pair of random variables X and Y is denoted
by:

Cov(X,Y).