The assumption of a quadratic utility function leads to mean variance analysis being optimum.

The variance of a random variable is defined as:

which is:

Since the expected value of the sum of random variables is the sum of the expected value, we have:

Since the expected value of a constant times a random variable is the constant times the expected value of the random variable, we can write the variance as:

and find that:

With a quadratic utility function of the form:

Taking the expected value we obtain:

Substituting our previously calculated relation we get:

and rearranging,