Proposition. If iXi, i=l,., n is a collection of independent Bernoulli random variables. then y defined via is binomially distributed Proposition. Let X be binomial with parameters n, p. Then EIX Proof. Exercise Now imagine that we are on a fishing expedition. For some reason we dip the fishing ole into the water 10 times for one minute at a time. The probability of catching a fish during any one dipping is p, independently of whether I caught a fish in any previous dipping. Then the total number of fish caught is a binomial variable But what if I dip 20 times for half a minute, or 40 times for 15 seconds etc. Assume that the probability of catching a fish during a half-minute dip is p/2 and similarly for shorter dips. What happens in the limit? As we take limits, let the expected total number of fish caught A= np be constant and let n -00.(It follows that p-0.) Then it turns out that the distribution of the total number of fish caught tends to the Poisson distribution with parameter A Definition. A random variable x is said to be poisson distributed if there is a real number \>0 such that P({X=k})=e- Proposition Suppose two independent random variables X and Y are PoissonProposition. If {Xi , i = 1, . . . , n} is a collection of independent Bernoulli random variables, then Y defined via Y = Xn i=1 Xi is binomially distributed. Proposition. Let X be binomial with parameters n, p. Then E[X] = np. Proof. Exercise. Now imagine that we are on a fishing expedition. For some reason we dip the fishing pole into the water 10 times for one minute at a time. The probability of catching a fish during any one dipping is p, independently of whether I caught a fish in any previous dipping. Then the total number of fish caught is a binomial variable. But what if I dip 20 times for half a minute, or 40 times for 15 seconds etc. Assume that the probability of catching a fish during a half–minute dip is p/2 and similarly for shorter dips. What happens in the limit? As we take limits, let the expected total number of fish caught λ = np be constant and let n → ∞. (It follows that p → 0.) Then it turns out that the distribution of the total number of fish caught tends to the Poisson distribution with parameter λ. Definition. A random variable X is said to be Poisson distributed if there is a real number λ ≥ 0 such that P({X = k}) = e −λ λ k k! . Proposition Suppose two independent random variables X and Y are Poisson 19