Slide 1Administrative …Random variablePMF/cDfRandom variableJoint distributionsJoint distributionsExpectationExpectationExpectationvarianceVariance, standard deviationCovariance, correlationCovariance, correlationchebyshevNextPROBABILITY AND STATISTICS IN COMPUTER SCIENCE AND SOFTWARE ENGINEERING Chapters 3: Discrete Random Variables and Their Distributions1ADMINISTRATIVE …The final will be on December 10 during class time (8:30-9:45am) in the Testing CenterWe are looking to create problem solving sessions – this would be a chance to work problems, do Q&A.Friday morning, 10:00-11:30am?Programming language: Matlab or R?R is free, Matlab may require a licenseBook uses MatlabThis is your choiceThe book is viewable online:2RANDOM VARIABLEA random variable is a function of an outcome, . It is a quantity that depends on chance. The domain of the function is the sample space We will assume that the range is a subset of the real numbers (could be integers, or an interval, etc.)Once an experiment is completed, and the outcome is known, the value of the random variable is determinedLook at Example 3.1 on page 40 The outcome is the result of flipping the coin three timesThe random variable is the number of heads Note that for each experiment (3 coin flips), we will get a value for X•C3PMF/CDFIf we look at the collection of probabilities related to X, we have the distribution of X. The function defined by is called the probability mass function, or pmf. The function defined by is called the cumulative distribution function, or cdf. The set of all possible values for X is called the support of the distribution F. See properties on page 41Note: Look at the pmf and cdf on page 42, and look at Example 3.2.•C4RANDOM VARIABLEWe have been looking at discrete random variables – the range is a finite (or countably infinite) set of points …It is also possible for the random variable to be continuous – in other words, it can take on all values in an interval (perhaps all real numbers)The concepts of pmf and cdf are now different – can’t use sumsThe book gives some example, but it fair to ask: Is there really such a thing as a continuous random variable in the real world?Aren’t all of our measurements limited by precision?Continuous random variables certainly exist in the abstract world …They will be useful to develop the theory that will explain the tools we use laterBy the way, random variables can in fact be mixed (combination of discrete and continuous)5JOINT DISTRIBUTIONSIf X and Y are random variables, then the pair (X, Y) is called a random vector. Its distribution is called a joint distribution of X and Y. The individual distributions of X and Y are called marginal distributions.We can define a joint probability mass function by the following:Section 3.2.1 shows the properties of these random vectors•C6JOINT DISTRIBUTIONSNote that just knowing the marginal probabilities does not give us enough information to know the joint distribution – we can’t generally compute if we only know the marginal probabilities …But there is one situation when this does workWe say that random variables X and Y are independent if This basically means the variables X and Y take on values independently of each other – the values are not related. For small joint distributions, it is easy to check independence – just use the above formula (see Example 3.6 on page 46)•C7EXPECTATIONWhen we have a distribution for a random variable (or vector), we can tell much about its properties from a couple of key parameters …The Expectation or expected value of a random variable X is its mean, the average value that it takes on.For a random variable that takes values ‘1’ and ‘2’ and has distribution , we would expect to get a ‘1’ half the time and a ‘2’ the other half of the timeThe expected value is Note that this is not actually a value that the random variable takes on!If we shift the probabilities, say , the expected value “shifts”; in this case it becomes 1.25. •C8EXPECTATIONWe can think of expectation like a center of gravityThink of weights being placed at the individual points on a line representing the values the random variable can take onThe expected value is the balancing point, the center of gravitySee diagram on page 49How to find the expected value? In general, we multiply each value that the random variable can take on by the probability for that value, and sum …•C9EXPECTATIONThere are some key properties of expectation that are shown on page 49 …Expectation is linear:For independent X and Y, we have•C10VARIANCEIf expectation is the average value that a random variable can be expected to take on, is there a measure for how much the random variable can vary from this average value?Consider our previous distribution () and another distribution (). Both have the same expected value: 1.50.But in some sense, the second distribution is “closer” to the expected value – it varies less …The variance of a random variable is the expected value of the square of the difference from each value to the mean …•C11VARIANCE, STANDARD DEVIATIONWhy squared? If not, we would always get ‘0’ by the linearity of expectationOther formulation:The standard deviation of X is defined by Units: Expectation is in the same units as X, variance is in units squared, standard deviation is in units•C12COVARIANCE, CORRELATIONWhen we have two random variable, X and Y, how can we measure how closely they are associated? In other words, do they increase/decrease together?The covariance is used to measure how well two random variables are associated:If large values of X correspond to large values Y (and similarly for small values), then the covariance is positive. If large X correspond to small Y and viceversa, then the covariance will be negativeCovariance close to zero means the random variable are uncorrelated …See pictures on page 51•C13COVARIANCE, CORRELATIONSince the size of the covariance may depend on the magnitude of the random variables, we often scale the result to get a correlation coefficient that belongs to the range [-1, 1] …Note that this is “dimensionless” – there are no unitsSee page 52-53 for a diagram and a listing of propertiesNote: Independent variables are always uncorrelated (because their covariance is zero)•C14CHEBYSHEVChebyshev’s inequality relates the expectation and the variance for a random variable …If the variance is
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