text: user1082: I found that several papers describe SNPs as a binomial distribution with the probability of "success" equals to minor allele frequency. 
  However, in my experiments, when I generate SNP array following this distribution, the simulation results behave very differently from the SNP array generated following the random mating procedures. I feel like there is some missing information about the description of a binomial distribution. 
  I wonder what will be a better mathematical description of the SNP array data? Besides a direct answer, any thoughts, or any suggestion of papers are also highly welcome. 
  EDIT:
  Thanks for these suggestions, after reading relevant materials, I think what I observe is about the Hardy-Weinberg equilibrium affected by evolutionary influences like mutation and migration. I will continue to look and list findings here if I find anything interesting. Please feel free to suggest some mathematical descriptions.  
  Also thanks for the suggestion of tools. They look very helpful, but I really hope to understand some mathematics behind it. 
  
  user920: Simulating genotypes with realistic correlation structures is indeed not so simple, and there's quite a few papers dedicated entirely to that (e.g. https://bmcgenet.biomedcentral.com/articles/10.1186/s12863-015-0173-4). Also, DEPICT (https://data.broadinstitute.org/mpg/depict/index.html) comes with a number of simulated GWASs to generate the nulls, so that's a simple way to obtain some ready-made fake data.
  On the other hand, indepedent SNPs should behave as draws from $Binomial(2, MAF)$. Note that the standard QC procedure of filtering out genotypes that deviate from Hardy-Weinberg equilibrium is just a goodness-of-fit test against $Binomial(2, \hat{MAF})$. If you employ this filter, some LD-pruning, and still get different results, consider posting more details about the discrepancies you see - that would be indeed unexpected.
  
original_text: 1082: <p>I found that several papers describe SNPs as a binomial distribution with the probability of "success" equals to minor allele frequency. </p>
  
  <p>However, in my experiments, when I generate SNP array following this distribution, the simulation results behave very differently from the SNP array generated following the random mating procedures. I feel like there is some missing information about the description of a binomial distribution. </p>
  
  <p>I wonder what will be a better mathematical description of the SNP array data? Besides a direct answer, any thoughts, or any suggestion of papers are also highly welcome. </p>
  
  <p><strong>EDIT:</strong></p>
  
  <p>Thanks for these suggestions, after reading relevant materials, I think what I observe is about the Hardy-Weinberg equilibrium affected by evolutionary influences like mutation and migration. I will continue to look and list findings here if I find anything interesting. Please feel free to suggest some mathematical descriptions.  </p>
  
  <p>Also thanks for the suggestion of tools. They look very helpful, but I really hope to understand some mathematics behind it. </p>
   920: <p>Simulating genotypes with realistic correlation structures is indeed not so simple, and there's quite a few papers dedicated entirely to that (e.g. <a href="https://bmcgenet.biomedcentral.com/articles/10.1186/s12863-015-0173-4" rel="noreferrer">https://bmcgenet.biomedcentral.com/articles/10.1186/s12863-015-0173-4</a>). Also, DEPICT (<a href="https://data.broadinstitute.org/mpg/depict/index.html" rel="noreferrer">https://data.broadinstitute.org/mpg/depict/index.html</a>) comes with a number of simulated GWASs to generate the nulls, so that's a simple way to obtain some ready-made fake data.</p>
  
  <p>On the other hand, indepedent SNPs <em>should</em> behave as draws from $Binomial(2, MAF)$. Note that the standard QC procedure of filtering out genotypes that deviate from Hardy-Weinberg equilibrium is just a goodness-of-fit test against $Binomial(2, \hat{MAF})$. If you employ this filter, some LD-pruning, and still get different results, consider posting more details about the discrepancies you see - that would be indeed unexpected.</p>
  