When I create a distribution by summing 5 different distributions and sample data from the summed distribution will I get normal distribution? This is a question regarding the central limit theorem. In my model, I have five sources of disturbances, each following a particular distribution. I sample the data from each and sum to determine the final disturbance. Will the distribution of the final disturbances be normal?
I read this sentence " In large problems, we can invoke the central limit theorem for all but the ﬁrst few completion times and use normal distributions as close approximations to the convolutions we need. " -Kenneth Baker
I am unable to justify it. Since only the means of the large sample follow normal distribution how can the random variable themselves form normal distribution when sampled?
This is a question on application of central limit theorem!
 A: As I understand the question:


*

*You have 5 independent random variables: eg. $u$, $v$, $w$, $x$, $y$.

*Random variable $z$ is the sum: $z = u + v + w + x + y$.

*You want to know if $z$ is normally distributed.


The answer is it depends (but from what you said, probably not)
Summing 5 independent random variables does not magically make the sum normally distributed.


*

*If $u$, $v$, $w$, $x$ and $y$ all are normally distributed, then their sum is normal.

*If any of $u$, $v$, etc... are not normally distributed:


*

*The sum is not going to be normal either (unless you're in a very peculiar case).

*As a practical matter, summing them will move things closer to the normal distribution, in some sense.



You can't invoke a classic central limit here because you only have 5 random variables. You can't answer this question in some abstract, general way. The answer will depend on what's the exact distribution of your disturbance terms. Why not do a QQPlot and see what it looks like?
Examples:
It could be really quite off (eg. this is a sum of a beta(.4..4), 3*gamma(1,.1), and 3 normals with stdev .05).

Or it could sorta be moving in the right direction... (eg. this is a sum of 5 IID uniform random variables)

But it's still off...

