Binomical vs poisson distribution in Simulations

[hagen]

Hey, I want to write a Computer Simulation in C++, which simulates the development of a DNA sequence with a probability to mutate x in one "generation". I do have a variable number (=n) of copies of this DNA. Now one might think, to simulate the mutation by simply:

sum(n*Poisson distributed random variable(x) )​

to get the number of mutated DNA copies. But this would be too slow.
So my question is, could I also just create a

binomically distributed random variable and multiply it by n * x​

to get the number of mutated DNA's? Or is this statistically incorrect?
If not, how might I set the Params for the Bin. dis.? Can I take 1 as a mean and multiply the result x or has the mean to be x? And how do I set / transform the variance of the distribution in a ratio to number of copies.
As you might probably have guessed, I'm a beginner in statistics, but i would be really grateful for any help. Thanks in advance,
Hagen

 

[radou]

Hm, if I remember correct, there is a link between the binomial and Poisson distribution.. Poisson's probability function is given with $$f(x)=\frac{\lambda^x}{x!}e^{-\lambda}$$. Now, I think you can put $$\lambda=mp$$, where the number of repetitions of a Bernoulli scheme experiment $$m \rightarrow \infty$$ and it's probability $$p \rightarrow 0$$.

 

[tacman]

In simulations, there are often situations where we need to model the probability of an event occurring. In your case, you are trying to simulate the development of a DNA sequence with a probability of mutation x in one generation. This type of event can be modeled using either a binomial or Poisson distribution.

A binomial distribution is used to model the number of successes in a fixed number of trials, where each trial has a constant probability of success. In your case, the number of successes would be the number of mutated DNA copies and the number of trials would be the number of copies of the DNA. This distribution is appropriate when the number of trials is fixed and the probability of success is constant.

On the other hand, a Poisson distribution is used to model the number of events occurring in a fixed time period, where the events occur independently and with a constant average rate. In your case, the events would be the mutations and the time period would be one generation. This distribution is appropriate when the time period is fixed and the events occur independently with a constant average rate.

In terms of simulation, both distributions can be used to model the probability of mutation in your DNA sequence. However, there are some differences between them that you should consider.

Firstly, the binomial distribution requires a fixed number of trials, which may not be appropriate in your case as the number of copies of the DNA can vary. On the other hand, the Poisson distribution does not have this limitation and can handle varying numbers of trials.

Secondly, the Poisson distribution assumes that the events occur independently and with a constant average rate, which may not be true in your case. If the probability of mutation varies depending on the specific DNA copy, then the binomial distribution may be a better choice.

In terms of setting the parameters for the binomial distribution, the mean and variance are determined by the number of trials (n) and the probability of success (x). The mean is equal to n * x and the variance is equal to n * x * (1-x). Therefore, if you want to simulate the number of mutated DNA copies, you can use a binomial distribution with a mean of n * x and a variance of n * x * (1-x).

In conclusion, both binomial and Poisson distributions can be used to model the probability of mutation in your DNA sequence simulation. The choice between them depends on the specific characteristics of your simulation and the assumptions you are willing