Characteristic Functions

1. Characteristic Functions 1. Bernoulli Distribution he Bernoulli distribution is a discrete distribution having two possible outcomes labelled by n=0 and n=1 in which n=1 ("success") occurs with probability p and ("failure") occurs with probability q=1-p, where 0 < p < 1. It therefore has probability function Examples which can also be written The corresponding distribution function is The characteristic function is

2. Characteristic Function • In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of nindependent yes/no experiments, each of which yields success with probabilityp. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial. In fact, when n = 1, then the binomial distribution is the Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance

3. ExampleA typical example is the following: assume 5% of the population is green-eyed. You pick 500 people randomly. The number of green-eyed people you pick is a random variableX which follows a binomial distribution with n = 500 and p = 0.05 (when picking the people with replacement Probability mass function In general, if the random variable X follows the binomial distribution with parameters n and p, we write X ~ B(n, p). The probability of getting exactly k successes is given by the probability massfunction: for k=0,1,2,...,n and where