1. Characteristic Functions Examples 1. Bernoulli Distribution he Bernoulli distribution is a discrete distribution having two possible outcomesdiscrete distribution 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 which can also be written The corresponding distribution function isdistribution function The characteristic function ischaracteristic function

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 n independent yes/no experiments, each of which yields success with probability p. 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 significanceprobability theorystatisticsprobability distributionindependentprobabilityBernoulli trial Bernoulli distributionbinomial test statistical significance

3. Example A 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 variable X which follows a binomial distribution with n = 500 and p = 0.05 (when picking the people with replacementrandom variable 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 mass function:probability mass function for k=0,1,2,...,n and where

4. Parameters number of trials (integer ) success probability (real)integerreal Support Probability mass functionProbability mass function (pmf) Cumulative distribution functionCumulative distribution function (cdf) Mean Median one of Mode Variance Skewness Excess Kurtosis Entropy mgf Char. func.