Definition 1: If a discrete random variable x has frequency function f(x) then the expected value of g(x) is defined as
Observation: The equivalent for a continuous random variable x is
This is the total area between the curve of the function h(x) and the x-axis where h(x) = f(x)g(x). For those of you familiar with calculus,
In order to avoid using calculus, we will restrict ourselves to the discrete case in the rest of this chapter, although all the results shown here for discrete random variables extend to continuous random variables. Click here for more details about how to extend the results presented here to continuous distributions.
Property 1: For any random variables x and y and constant c
- E[c] = c
- E[cg(x)] = cE[g(x)]
- E[g(x) + h(x)] = E[g(x)] + E[h(x)]
- E[xy] = E[x] ∙ E[y] if x and y are independent
Proof: (a) – (c) are simple consequence of Definition 1. (d) is a consequence of Property 2 of Discrete Distributions.
Definition 2: If a random variable x has frequency function f(x) then the (population) mean μ of f(x) is defined as
Here the function g(x) in Definition 1 is the identity function g(x) = x.
The (population) variance σ2 is defined as
Property 2: The variance can also be expressed as
Proof: By Property 1,
Property 3: For any random variable x and constants a and b
Proof: The first assertion is a consequence of Property 1, namely:
For the second assertion, by Property 1 and 2, we have:
Observation: It follows from Property 3 that for any constant b, Mean(b) = b and Var(b) = 0.
Proof: The first assertion follows from Property 1:
For the second assertion, by Property 1 and 2
But by Property 1d, E[xy] – E[x]E[y] = 0 since x and y are independent, and so
Definition 3: For any random variable x with mean μ and standard deviation σ, the standardization z of x is defined by
Property 5: The standardization of any random variable has mean 0 and variance 1.
by Property 3 the mean of z is
Excel Function: Excel provides the following function for calculating the value of z from x, μ and σ:
STANDARDIZE(x, μ, σ) = (x – μ) / σ
Definition 4: The nth moment around the mean is defined as
Click here for more advanced information about moments and related subjects.
Observation: It follows from Definitions 2 and 4 that the variance can be expressed as
In Symmetry and Kurtosis we define the skew and kurtosis of a sample. We now define the population equivalents of these concepts as follows.
Definition 5: The (population) skewness is defined as
The (population) kurtosis is defined as
Observation: The 3 in the kurtosis definition is the value of μ4/σ4 for the normal distribution function (see Normal Distribution). Thus the kurtosis of the normal distribution function is 0.