Discrete distributions: uniform, binomial and Poisson

In short

• Discrete uniform: n equally likely outcomes, each with probability 1/n. For consecutive integers from a to b, mean $(a+b)/2$ and variance $[(b-a+1)^2-1]/12$.
• Bernoulli: one trial, success with probability p. Binomial: the number of successes in n independent Bernoulli trials, with mean np and variance np(1 − p).
• The binomial model of stock prices moves the price up by a factor u or down by d each step, so after i ups and j downs $S = S_0 u^i d^j$.
• With large n and both np and n(1 − p) above 5, a binomial is well approximated by the normal $N(np,\, np(1-p))$.
• Poisson: the number of events in an interval when events arrive independently at a constant rate λ. Mean = variance = λ; skewness $1/\sqrt{\lambda}$.
• In default-intensity models $P(\text{no default by } t) = e^{-\lambda t}$, $P(\text{default by } t) = 1 - e^{-\lambda t}$, and the expected time to default is $1/\lambda$.