CIS2460 Statistics Tutorial Part 3

Statistical TutorialStatistical Concepts Part 3(Chapter 5)Compiled and Presented byDiscrete DistributionsAlana CordickDiscrete Distributions Bernoulli Trials• The following are discrete distributions: • Consider an experiment consisting of ntrials• Bernoulli trials and Bernoulli distribution• Each can be a success or failure• Binomial distribution• Where X = 1 if the jth experiment is a success• Geometric and negative binomial distribution j• X = 0 if the jth experiment is a failurej• Poisson distribution1Bernoulli Distribution Bernoulli Distribution• One trial of the Bernoulli Distribution is: • Expected ValueSuccess• E(X ) = 0 * q + 1 * p = pjp, x =1, j=1,2,...,njp (x )= p(x )= 1 p= q, x = 0,j=1,2,...,nj j j j• Variance0, otherwiseFailure2 2 2• V(X ) = [(0 *q) + (1 *p)] – p = p (1-p) = p qjBernoulli Process Binomial Distribution• The random variable X that denotes the number of successes inn Bernoulli trials has a binomial distribution p(x)• n Bernoulli trial are a Bernoulli process IFnx n x p q , x= 0,1,2,...,n1. The trails are independentp(x)=x• p(x ,x ,…, x ) = p (x ) p (x ) … p (x )0, otherwise1 2 n 1 1 2 2 n nThe number of2. Each trial has only 2 possible outcomes Probability thatoutcomes having thethere arerequired number ofx successes and• Successsuccesses and(n-x) failuresfailures• Failure• X is a sum of n independent ...