Sampling - fitting of sample Does sample size influence the underlying distribution of population, i.e., the bigger the sample you generate the more it resembles whatever distribution of the population it's drawn from?
What would you say are the key factors for measuring the quality of a given sample with regards to its fitting to the distribution of its population?
I'm currently studying discrete random variables, eg. the geometric distribution.
Thank you.
 A: 
Does sample size influence the underlying distribution of population

The size of the sample doesn't change the population; indeed the question suggests that you've misunderstood one of the terms (or perhaps didn't quite ask what you meant). Did you mean to ask something else?

the bigger the sample you generate the more it resembles whatever distribution of the population it's drawn from?

This is a very different question to the one you asked in the first part of the sentence. 
Yes, under random sampling, larger samples on average more closely resemble the population than smaller samples - the empirical cdf, $\hat F$ approaches the cdf, $F$. This page mentions several results relating to the convergence:
http://en.wikipedia.org/wiki/Empirical_distribution_function#Asymptotic_properties

What would you say are the key factors for measuring the quality of a given sample with regards to its fitting to the distribution of its population?

I'm not sure I quite follow what this question is getting at. If you define some quantity you're interested in measuring (such as, for example, a Kolmogorov-Smirnov type distance - the infinity norm), then you may be able to compute useful properties (such as the distribution of $\sqrt n D$), but what is 'key' depends on what you're doing/trying to find out. Take a look at the panoply of results at that link above.
Simulation is a useful tool for investigating the effect of increasing $n$, but to see the properties in the limit really requires computation of asymptotic properties; the link is a useful place to start investigating some of them.
A: @Glen_b covered your first question pretty well. So my answer just pertains to your desire to assess sample quality or, probably a better term, reliability. Since in a real situation you will not know the underlying distribution, you will not be able to directly assess or simulate the sampling distribution of various sample sizes. However, you can get an idea of how accurate your sample is by forming nonparametric confidence intervals for key percentiles based on order statsitics (e.g., 10th, 20th, 30th etc) and seeing how wide they are and if they overlap. 
If you are willing to assume a distribution for the population as a guess, then I would go with direct Monte Carlo simulation at various sample sizes to see the resulting sampling distribution. The arbiter of quality will usually be the length of the interval required to cover X% of the sampling distribution, which will be your confidnece interval length. You can see how many samples you need to get your desired interval length and confidence level. Of course, I'd try varying your assumed distribution to verify that your sample size is robust against misspecificaion of your assumed population distribution.
