0
$\begingroup$

Snowflakes are falling at an average rate of 10 flakes per square inch per minute. Calculate the probability that a 2 square-inch region has no snow flakes in a given 5 second time interval.

How should I know to use Poisson distribution in this case? What are the events to be modeled?

  • 1
    We can model the situation by imagining $n$ snowflakes, with $n$ a million or more, each independently having probability $p$ of hitting our square inch bald patch, where $np=10$. In principle we have a binomial distribution. In practice, aince $n$ is very large and $np$ is modest, we can approximate the binomial distribution by using the Poisson.2012-10-08
  • 0
    Poisson approximates Binomial especially well if $p=\varphi(n)$,e.g. $p=\frac{1}{n}$, then $\lambda=np=1$2012-10-09

1 Answers 1

2

Poisson PDF is $\frac{e^{-\lambda}{\lambda}^{k}}{k!}$ with $k=0, \lambda=10$. But in your case there are two more parameters: space and time, so the probability becomes $$ P(\textbf{ no flakes})=\left. \frac{e^{-\lambda t s}{(\lambda t s)}^{k}}{k!} \right|_{t=\frac{1}{12},\ s=2,\ k=0}=e^{-\frac{5}{3}} $$