Expectation

Expected Value

Definition 1: If a discrete random variable x has frequency function f(x) then the expected value of the function g(x) is defined as

The equivalent for a continuous random variable x is

where f(x) is the probability density function. 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

1. E[c] = c
2. E[cg(x)] = cE[g(x)]
3. E[g(x) + h(x)] = E[g(x)] + E[h(x)]
4. E[xy] = E[x] ∙ E[y] if x and y are independent

Proof: (a) – (c) are simple consequences of Definition 1. (d) is a consequence of Property 2 of Discrete Distributions.

Population Mean and Variance

Definition 2: If a random variable x has probability density 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 1c, namely

For the second assertion, by Properties 1 and 2, we have

It follows from Property 3 that for any constant b, Mean(b) = b and Var(b) = 0.

Property 4: For random variables x and y

If in addition, x and y are independent, then

Proof: The first assertion follows from Property 1:

For the second assertion, by Properties 1 and 2

But by Property 1d, E[xy] – E[x]E[y] = 0 since x and y are independent, and so

Standardization

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 a mean of 0 and a variance of 1.

Proof: Since

by Property 3 the mean of z is

and the variance 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

Real Statistics Function: The Real Statistics Resource Pack provides the following array function for the array or cell range R1.

XSTANDARDIZE(R1) = STANDARDIZE(R1,AVERAGE(R1),STDEV.S(R1))

Moments

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

Skewness and Kurtosis

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

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.

Reference

Wikipedia (2012) Expected value
https://en.wikipedia.org/wiki/Expected_value

Penn State (2021) Mathematical expectation. Introduction to probability theory
https://online.stat.psu.edu/stat414/lesson/8