Calculating Variance of Independent Random Variables: A Simple Guide

[ericm1234]

1. I know var(x)=E(x^2)-E(x)^2; is there a repeated way to use this to attain var(x^2)? Or how in general, without resorting to integration, can I calculate it?

2. We typically deal with "i.i.d random variables X_i" and do things like find var(X) given E(X^2) etc..it never occurred to me until now, but if the X's are "independent" then why is E(x^2) not equal to E(x)*E(X)?? (the answer I'm awaiting will probably be obvious, though I can't figure this out right now)

 

[Simon Bridge]

1. put Y=X^2 and find var(y).
2. do the math and see.

 

[ericm1234]

Doesn't help because I then need to find E(x^4); I'm dealing with a continuous function, hence my question about trying to avoid integrating.

"do the math" doesn't help my understanding. The term 'independent' is used in regards to a series of random variables X from the same distribution and yet the definition of independence doesn't apply here as far as E(X^2)+E(X)*E(X) being true.

 

[ericm1234]

I can make up an example to show it's obviously not true; I'm asking for an explanation of the use of the word independent in this context.

 

[Simon Bridge]

Doesn't help because I then need to find E(x^4); I'm dealing with a continuous function, hence my question about trying to avoid integrating.

I suspect you have misunderstood - you asked a question - I pointed you to the path where you are most likely to be able to find the answer. This objection/protest suggests to me that you found the answer - thus: it did help ;)

But it is possible that I misunderstood - perhaps you could rephrase your question?

i'm asking for an explanation of the use of the word independent in this context.

I'm afraid I can only answer the questions you write down. You wrote:

if the X's are "independent" then why is E(x^2) not equal to E(x)*E(X)?

... and that was the question I answered.
You are correct that a specific example will not suffice - have to work harder than that.
Try rewriting the question as a mathematical statement you have to prove/disprove. i.e. $E[X^2]=[E[X]]^2$ ... but expand it to the definitions.

Now to your new question:
The word "independent" is a label for a set of mathematical properties that you can best understand by doing the math. Turn the thought around: what is it about the mathematical property of "independent", in this context, that leads you to think E(x^2) should have that form?

 

[ericm1234]

It is by the definition of "independence" in statistics that E(X*Y)=E(X)*E(Y). If two X's are independent should not E(X^2)=E(X)*E(X) from this context?
A bunch of iid X's from say, a normal distribution are independent with each other and yet do not fit the above definition of independence. Please explain/point out where my simplistic reasoning has failed.
It was awhile ago that I dealt with my stats material; I am looking for a straight forward explanation of why these two uses of the word "independent" do not mesh; I am not looking for an exercise.

 

[statdad]

The reason E(X^2)=E(X)*E(X) does not work is that X refers not to two different random variables, identically distributed but independent, but to a single random variable.
You don't need to carry out the integrations to see mathematically why the two expressions E(X^2) and E(X)*E(X) are not the same, just do this:

Step 1: Write out the integral that gives E(X^2)

Step 2: Write out the product of the two integrals that give E(X) * E(x)

and think about why the two expressions are not the same

 

[Simon Bridge]

Thanks statdad - that's pretty much what I've been trying to get ericm1234 to do ;)
Spelling it out is the next step.

@ericm1234:
Since you resist writing down any actual math... try thinking about it this way:
if you have two distributions Y and Z, but both of them depend on a third distribution X, then are Y and Z independent of each other? i.e. is X independent of itself?

But seriously, you must get used to thinking in terms of the actual math.