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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:

x f(x)
2 0.01
3 0.25
4 0.4
5 0.3
6 0.04

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 σg(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).

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