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