Slide 1Distributions: DiscreteDistributions: bernoulliDistributions: BinomialDistributions: BinomialDistributions: GeometricDistributions: geometricDistributions: negative binomialDistributions: negative binomialDistributions: poissonDistributions: poissonDistributions: summaryPROBABILITY AND STATISTICS IN COMPUTER SCIENCE AND SOFTWARE ENGINEERING Chapters 3: Discrete Random Variables and Their Distributions1DISTRIBUTIONS: DISCRETEIn this section we will review the basic discrete distributions that are commonly used when working with discrete random variablesEach has unique properties and usesThese are particularly useful for modeling and simulationWe will soon see that much of statistical inference is based upon estimating the parameters of (and thus discovering) the underlying distribution for a population that has been sampled2DISTRIBUTIONS: BERNOULLIA random variable with two possible values, 0 and 1, is called a Bernoulli variable. Its distribution is called a Bernoulli Distribution, and an experiment with a binary outcome is called a Bernoulli trial.Heads or tails, good or defective parts, transmitted or lost signals, malignant or benign, etc.Generally two outcomes: Success or FailureProperties listed on page 58Note that there is whole family of Bernoulli Distributions – they each depend upon the parameter p3DISTRIBUTIONS: BINOMIALA variable described as the number of successes in a sequence of independent Bernoulli trials has a Binomial distribution. The parameters are n (the number of trials) and p the probability of success for each trial.Flip a coin 10 times, what is the probability of getting 3 heads?Here The pmf for this distribution is shown on page 58 – note the term , which accounts for the different orderings that may occur in the outcomesGenerally for this distribution, probabilities are calculated using a table like the one shown in A2 (page 412)•>4DISTRIBUTIONS: BINOMIALNote the table in A2 is a cdf … how to use this to calculate a probability for a given x?Since x is an integer, we can just use the following:,Where the values for F(x) can be found from the table in A2See example 3.16 on page 59 …The discrete variable for a Binomial distribution can be thought of as the sum of n discrete Bernoulli variables, which take on values 0 or 1 …Using the linear properties of expectation and variance, we can then easily derive the properties of the Binomial distribution (see page 59)•>5DISTRIBUTIONS: GEOMETRICSuppose we conduct a sequence of Bernoulli trials – how many do we need to conduct before we get our first “success”?This can be defined as a random (discrete) variable, and has a distribution …The number of Bernoulli trials needed to get the first success has Geometric distributionThe number of coin flips needed for a tails to appear …This random variable can take any value from 1 to infinityBut if we think of the coin flip example, you would expect that the pmf “trails off to zero” as x gets larger6DISTRIBUTIONS: GEOMETRICSince each trial is Bernoulli, and we only have success on the last one, it is easy to see that the formula for the pmf for a Geometric Distribution is given byNote that we do not need to consider “orderings” as in the Binomial Distribution Properties of this distribution are shown on page 61•>7DISTRIBUTIONS: NEGATIVE BINOMIALThis distribution is an extension of the Geometric distributionIn a sequence of Bernoulli trials, the number of trials needed to obtain k successes has Negative Binomial distributionFor the Binomial distribution, we count the number of successes in n trials. For the Negative Binomial, we determine the probability that the success will occur on the trial.What is the probability of getting a fourth heads in a sequence of ten coin flips?It is the probability of getting three heads in the first nine flips (in any order) and then getting a heads on the tenth flip•>8DISTRIBUTIONS: NEGATIVE BINOMIALThis distribution has two parameters: k (which is the number of successes) and p (which is the probability of success for each trial)For , the Negative Binomial just becomes the Geometric distributionIn fact, the Negative Binomial variable can be thought of as the sum of Geometric random variables …From here, is it easy to derive the properties of this distribution(see page 63)•>9DISTRIBUTIONS: POISSONThis distribution is used to model rare events, and in particular interarrival times of these eventsThe number of rare events occurring during a fixed period of time has Poisson distributionNote this is different than Binomial, which deals with a fixed number of Bernoulli trials (and determines the probability of a number of successes occurring during those trials)For Poisson, we have only one parameter: , which represents the frequency or average number of eventsThe pmf for this distribution is given on page 65, and the properties are also derived there•>10DISTRIBUTIONS: POISSONNote that while Poisson and Binomial are different distributions, they are related …Poisson distribution can be used to approximate the Binomial distribution when n is large and p is smallNote that table A2 only goes up to Generally, if and , the Binomial pmf P(x) for a random variable x can be approximated by the Poisson pmf with The reason for this is shown on page 66; basically, the binomial expression approaches the exponential expression as n gets large, p gets small, and np approaches Can also do this for larger p; just use in the calculation (see example 3.24)•>11DISTRIBUTIONS: SUMMARY12DistributionUse ParametersE(X) Var(X)Bernoulli Coin flip p pBinomial How many successes in n trials?n, p npGeometric How many trials until first success?pNegative BinomialHow many trials needed to obtain k successes?k, pPoisson Number of rare events occurring in a fixed period of timeDistributionUse ParametersE(X) Var(X)Bernoulli Coin flip p pBinomial How many successes in n trials?n, p npGeometric How many trials until first success?pNegative BinomialHow many trials needed to obtain k successes?k, pPoisson Number of rare events occurring in a fixed period of
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