Need help for Normal Distribution

frekiwang SG

I need a real-life(or science-related) example which illustrates the fact that the sum/difference of two dependent normal variables may not follow normal distribution.

Please help, thanks

wee_ws SG

Hi frekiwang,

Someone constructed a theoretical example:

http://www.mathforum.org/kb/message.jspa?messageID=1543275&tstart=0

Another example by Holton:

http://www.value-at-risk.net/content/Holton_3.pdf

Please see page 138 or page 33.

It seems that the real-life application is in risk management (of portfolios related to investments and insurances).

Here is another reference:

http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5516211

In this case, it is motivated by examples from quality and reliability engineering.

Thanks.

Cheers,
Wen Shih

frekiwang SG

thanks for your help, but I still need a real-life or science-related example.

wee_ws SG

Hi,

I believe it is not easy to find specific examples, rather it is easier to let students appreciate the possible applications in general contexts (e.g., risk management in finance and insurance, quality and reliability engineering).

The simple theoretical 'toy' example serves to illustrate concretely the idea.

Thanks.

Cheers,
Wen Shih

SBS2601D SG

The assumption that the sum/difference of 2 dependant variables will be normal requires the assumption that there are no other variables that affect the endogeneous variable and are also correlated with the exogeneous variable.

That means if changes in X causes changes in Y, and there is a variable Z that affects Y but is also correlated to X, then the sampling distribution of the sum/difference of X and Y will not be normal.

An easier way would be that if the causality runs both directions, ie X affects Y and Y affects X. Then the sum/difference will also not be normally distributed.

Now that the tools have been given, it would be your job to relate an example because there's plenty of them around us.

frekiwang SG

An easier way would be that if the causality runs both directions, ie X affects Y and Y affects X. Then the sum/difference will also not be normally distributed.

Now that the tools have been given, it would be your job to relate an example because there's plenty of them around us.

Thanks.

I know how to prove it mathematically at U level and I can easily construct a few theoretic models to show it, but here I need a specific example to illustrate this fact to JC students. Your first paragraph to them will be some kind of alien language already, they will not understand why there must be no other variables that affect the 'endogeneous' (alien to them again) variable in order to make the sum/different follow normal.

What I need is something like this:

In the context of a real-life example, X denotes ... Y denotes...., when X and Y both follow normal distribution, it is obvious that X+Y(or X-Y) does not follow normal, and the reason is that X and Y here are dependent. 'Obvious' means JC students should be able to see it without doing any mathematical proof/reasoning.

SBS2601D SG

I understand what you mean. Unfortunately even after type, backspace, type, backspace, repeat, I couldn't simplify it any further.

Let's see. The examples I deal with often include the demand/supply functions determining prices and quantity. Because these 2 variables influence one another both ways (simultaneous causality), the residual of the regression would be non-normal in distribution.

Hope that helps!

Edit: Out of curiosity, what are you trying to prove to the JC students regarding this?

frekiwang SG

I understand what you mean. Unfortunately even after type, backspace, type, backspace, repeat, I couldn't simplify it any further.

Hope that helps!

Edit: Out of curiosity, what are you trying to prove to the JC students regarding this?

I mean X+Y or X-Y only please, so in your example, are you implying demand+/-supply does not follow normal or price+quantity does not follow normal. I suppose the latter is not valid as it meaningless to sum up price and quanty. Demand - Supply makes sense, but how to explain to JC students that the result of 'demand - supply' obviously does not follow normal.

I am trying to prove, or rather, to use a real-life example to show the JC students that if X~N(u, a^2), Y~N(v, b^2), the condition of indepence is needed to conclude X+/-Y still follows normal.

SBS2601D SG

I meant price-quantity, but of course its relevance is limited to JC students because they would not have learnt regression and its uses.

I can only help this far because I use such statistical tools for what I do at the moment, which is beyond JC level, and which I wasn't taught at that level also (probably for a good reason).

Darkness_hacker99 SG

The topic in this thread is something that I'm not good with. I read and read this post and I still couldn't understand much about this, and it makes me pretty upset because I can't offer my help. It makes me recall a quote from Albert Einstein that I would like to share it with you guys.

He once said that if he had 1 hour to save the world, he would spend 55 minutes defining the problems and only 5 minutes finding the solutions.

wee_ws SG

Hi,

This advice is a good one, thanks for sharing!

Cheers,
Wen Shih