How can you prove that a random variable follows the Poisson distribution?

You could draw a Probability Plot: "The probability plot ... is a graphical technique for assessing whether or not a data set follows a given distribution such as the normal or Weibull. "The data are plotted against a theoretical distribution in such a way that the points should form approximately a straight line. Departures from this straight line indicate departures from the specified distribution." Source: Online Engineering Statistics Handbook http://www.itl.nist.gov/div898/handbook/eda/section3/probplot.htm