**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*[*x*y] =*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** *σ*^{2} 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.

**Property 4**:

Proof: The first assertion follows from Property 1:

For the second assertion, by Property 1 and 2

But by Property 1d, *E*[*x*y] – *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 *μ*_{4}/*σ*^{4} for the normal distribution function (see Normal Distribution). Thus the kurtosis of the normal distribution function is 0.

I was looking for the information on moments, the link given above is not working. Could please provide with the working link. Your explanation and derivation is very intuitive.

Thanks Regards

Harsha

Harsha,

Sorry about that. I have just repaired the link. In any case, here is the link that I believe that you are looking for

http://www.real-statistics.com/general-properties-of-distributions/advanced-properties-of-distributions/properties-probability-distributions-detailed/

Charles

Hi,

a little error here

Sigma/Sigma^2 = 1

To correct, Signa^2/Sigma^2 = 1

This web site is awesome. It’s really useful providing many excel add-in functions.

Thanks, Sir

Hi Mobb,

I am very pleased that you like the website.

I also appreciate your identifying the typo. I have now corrected the error on the referenced webpage. Thanks for catching it.

Charles