gle independent trial is q, the binomial 2.2 Exponential, Poisson, and binomial distributions Figure 2.4 Plots of the binomial or Bernoulli distribution corresponding to parameters q = 0.1, 0.25, 0.5 with n = 10 mean (open symbols corresponding to the case q = 0.1). PDF is defined according to: (2.9) Such a definition makes sense, considering that the probability of obtaining k indepen- dent successes is qk and the probability of failure for the remaining n - k events is (1 - q)n-k The product of the two is the joint probability of the group of measurements, as observed in any specific order. The combinatorial factor CK is the number of possible orders for successive success or failure measurements. The mean and variance of the binomial distribution is N = nq and o2=N(1-q), = - respectively. Figure 2.4 shows plots of the binomial PDF for n = 10 and different values of q. We observe that the PDF is asymmetric except when q = 0.5, in which case the peak value is exactly centered about N. A physical illustration of the Bernoulli distribution is the passing, one at a time, of a stream of individual particles (like light quanta or photons) through a piece of absorbing material. These particles may either successfully pass through the material (success, probability q), or absorbed by it (failure, probability 1 - q). Assuming an ideal particle counter, the probability of counting k particles after the stream has passed though the material is effectively p(k), as defined by Eq. (2.9). See also Section 2.5 in the bean machine example. We note that the binomial distribution is defined over the finite random-variable interval 0 k < N. Since there is no restriction on the size of the sampling space defined by n, the PDF can be extended to the limit n 8. It can be shown that in this infinite limit, the binomial PDF is equal to the Poisson PDF previously described. It can also be shown that, for large mean numbers N, the Poisson (or binomial) PDF asymptotically converges towards the Gaussian or normal distribution, which is de