title: How to approximate a Poisson distribution $P(n,lambda)$
body: <p>I have to implement an algorithm to evaluate a sum of poisson functions, each one with multiplying constants:</p>
  
  <p><img src="https://i.stack.imgur.com/gPc3x.gif" alt="Required Sum"></p>
  
  <p>Where C(k) are positive constants&lt;1, cut is a cutoff because in principle the sum should take infinite numbers of k, and lambda is a number that may vary in my case from 20 to 100.  I've tried a straight forward implementation in my code:</p>
  
  <pre><code>    #include&lt;quadmath.h&gt;
     ... //Definitions of lambda and c[k]...
      long double sum=0;
      for(int k=0;k&lt;cutoff;++k)
      {
       sum=sum + c[k] powq(lambda,k)/tgamma(k+1)* (1.0/expq(lambda));
      } 
  </code></pre>
  
  <p>But I am not quite satisfied. I've searched on "Numerical recipes" for a good approach to evalutation of a poisson distribution, but I didn't find anything about that. 
  Are there better ways to do this?</p>
  
  <hr>
  
  <p>Edit: to be clear, I'm looking for the most precise way to approximate the probaility of large events, given a poisson distribution, without computing awkward (lambda^k)/k! Factors!</p>