Slide 1Distributions: summaryNegative BinomialProbability densityProbability densityProbability densityProbability densityDistributions: uniformDistributions: uniformDistributions: exponentialDistributions: exponentialPROBABILITY AND STATISTICS IN COMPUTER SCIENCE AND SOFTWARE ENGINEERING Chapters 4: Continuous Distributions1DISTRIBUTIONS: SUMMARY2DistributionUse 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 timeNEGATIVE BINOMIALThe Negative Binomial distribution was introduced last time as a probability distribution which provided the pmf for the number o Bernoulli trials needed to obtain k successesRecall the Geometric distribution talked about the probability that the first success occurred on the k-th trial, and we considered the Negative Binomial to be a series of Geometric variablesWe found properties (listed on page 63), but noted that there was no table for the distribution – how to compute?We can see in example 3.21 on page 64 that problems involving the Negative Binomial can usually be re-phrased as Binomial problems3PROBABILITY DENSITYThe first observation is that for continuous variables, there is no such thing as probability mass function (pmf), because for any variable xThis is because for discrete random variables, we have a finite or countably infinite set of values, so the expression makes sense; not the case for continuous x, since this variable is uncountably infiniteHowever the cumulative density function (cdf) does make sense, and can be defined by Again, F(x) is a non-decreasing function that ranges from 0 to 1For discrete random variables, the graph of F(x) has jumps (at each discrete point)For continuous variables, it is a continuous curve (since )•@4PROBABILITY DENSITYFor most of the commonly known distributions, F(x) is differentiable (has a derivative)We can then define a probability density function (pdf) which takes the place of the pmf …The probability density function (pdf) is the derivative of the cdf, namely . A distribution is called continuous if is has a densityWhile saying does not make sense for a continuous random variable, we can use f(x) to compute the probability that the random variable x lies in a range, i.e. …See page 76 for more details. The comparison between discrete and continuous random variables is shown in Table 4.1 on page 77•@5PROBABILITY DENSITYMany of the properties of discrete random variables have analogous properties in the continuous case …In particular, we can joint cumulative distribution functions for multiple continuous random variables, and (using partial derivatives) define joint density functions too …Concepts for marginal distributions and independence also carry overThese are shown in Table 4.2 on page 786PROBABILITY DENSITYWhat about expectation and variance?These are critical concepts that also have analogous definitions for continuous random variablesRecall that for discrete random variables, expectation was thought of as the balancing point for all of the weights (representing the probability masses) placed at each discrete variable …For continuous random variables, each point has zero mass, but we can still calculate a center of gravity for the area under the probability mass function …From this we get the formula for E(x) of a continuous variable x; similar derivations hold for variance and standard variation (see page 79)7DISTRIBUTIONS: UNIFORMThe simplest continuous random variable density function is the uniform distributionIn this distribution, all values of x in a range are given equal preference – a uniform distributionThe pdf for this distribution is given by Note the area under this curve is equal to 1While does not make sense for this continuous variable, we can (easily) compute for any t, h – and it turns out this probability is independent of tThis makes sense since the distribution is uniform – see pages 80-81•@8DISTRIBUTIONS: UNIFORMOne particular uniform distribution is important – the Standard Uniform DistributionThis is the uniform distribution where (i.e. the interval is just [0, 1])We can transform any continuous uniform random variable Y to a standard uniform random variable X with the transformationLikewise, we can transform a standard uniform variable to a uniform variable on any interval [a, b] (see page 81)We can easily compute the expectation and variance for these distributions – see page 81-82. Note the expectation makes perfectly good sense – it is the middle of the interval [a,b], which is where you would expect the center of gravity to be•@9DISTRIBUTIONS: EXPONENTIALExponential distribution is often used to model time – in particular, inter-arrival times, waiting times, etc. It is related to the Poisson distribution (which was used to provide the probability mass function for the number of rare events occurring in a time period) Poisson provides the number of events, exponential provides the inter-arrival timesThis distribution has one parameter (similar to Poisson): , which represents the frequency of events, measured in .We can see on page 82 the formula for the pdf, cdf, expectation and variance for this distribution•@10DISTRIBUTIONS: EXPONENTIALThe pdf for the exponential distribution decays exponentially as x goes to infinity, meaning longer inter-arrival times are less likely than shorter onesPage 83 also shows how, if we assume that the rare events are Poisson distributed, we can see why the inter-arrival time is exponentialEssentially, we look at the (Poisson) probability of no rare events occurring before time t, and when we look at the cdf for this probability we get the exponential distribution. Since t is not discrete, this is in fact a continuous distributionThis distribution also has the unusual property of being memoryless – this means the amount of time we have already waited does not affect the future waiting time (it is “forgotten”) – see derivation on page
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