Characteristic FunctionsCharacteristic FunctionExample 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 replacementSlide 4Characteristic FunctionsExamples1. Bernoulli Distribution he Bernoulli distribution is a discrete distribution having two possible outcomeslabelled 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 is The characteristic function isCharacteristic 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 significanceExampleA 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 replacementProbability mass functionIn 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: for k=0,1,2,...,n and whereParameters number of trials (integer) success probability (real)Support Probability mass function (pmf) Cumulative distribution function (cdf) Mean Medianone of Mode Variance Skewness Excess Kurtosis Entropy mgf Char.