State the probability functions

Topic: Functions of random variables and limit theorems · more questions on this topic
‹ A normal law with mean 2 Partial sums of standard normals ›

  1. State the probability mass function of the binomial, geometric and Poisson distributions, in the parameterisation of the common distributions page. The geometric law starts at 1.

  2. State the densities of the uniform distribution on [a,b], the exponential distribution with rate \lambda, and the standard normal distribution.

  3. For each law in (a) and (b), name an experiment that produces it.

  1. Binomial, x=0,\ldots,n: \binom{n}{x}p^x(1-p)^{n-x}. Geometric, x=1,2,\ldots: p(1-p)^{x-1}. Poisson, x=0,1,\ldots: e^{-\lambda}\lambda^x/x!.

  2. Uniform on [a,b]: density 1/(b-a) on that interval. Exponential rate \lambda: density \lambda e^{-\lambda x} for x>0. Standard normal: density (2\pi)^{-1/2}\exp(-x^2/2) on \mathbb{R}.

  3. The binomial count is the number of successes in n independent trials. The geometric count is the trial number of the first success. The Poisson count is the limit of a binomial count with mean held fixed while n grows, and it models rare events in a fixed window. The uniform law is a point drawn so that probability equals length. The exponential law is a waiting time that does not age. The standard normal law is the limit of a centred and scaled sum of independent copies with mean 0 and variance 1.