VarianceBasic IdeaBasic Idea (continued)Computing VarianceVariance of a Finite Random VariableStandard DeviationExampleSpecial CaseContinuous Random VariableSlide 10Example #2SamplesExamplesSummary ChartSample Mean as a R.V.Focus on the ProjectStandardizationSlide 18VarianceFall 2003, Math 115BBasic IdeaTables of values and graphs of the p.m.f.’s of the finite random variables, X and Y, are given in the sheet computations of the file variance example.xls.Let X be the random variable that gives the value of the die when a 10-sided die is rolled once. The sides of the die are labeled 1 through 5.Let Y be the random variable that gives the average of the 2 sides of 10-sided die when it is rolled twice. The sides of the die are labeled 1 through 5.Basic Idea (continued)The mean of a distribution is a measure of center.Although the means are the same, the concentration of the observations for each random variable is quite different.The mean of the random variable does not accurately reflect where the probabilities lieWe need a parameter that measures the spread or dispersion of the possible values.What is a parameter?Computing VarianceCompute x - µX for all possible values of X.What does this difference represent?What happens when we sum all of these differences up?Compute the sum of the squares of the deviations of the possible values from the mean.Does not take into account the relative likelihood of the possible values of X.Can weight each term with the probability of getting the value X.Variance of a Finite Random VariableWe can now calculate (x - µX)2 * fX(x)Each value of the p.m.f weights each term with the probability of obtaining the value x.This weighted sum is called the variance of X, where X is a finite random variableDenoted by V(X)V(X) = Σ (x - µX)2 * fX(x), for all possible values of xWhat does the variance tell us?Standard Deviation V(X) is one measure of dispersion. The square root of V(X) is the preferred measure of dispersion. Denoted by σX – called the standard deviationMeasured in the same units as X and µX Interpreted as “typical amount” by which a value of X will differ from µX σX ≥ 0ExampleThe p.m.f of X, the number of Americans in a sample of size 4 that have at least 1 credit card is given to the right:Find V(X) and σXx fX(x)0 0.003321 0.041902 0.199623 0.421424 0.33374Special CaseWhat happens if X is a binomial random variable?Then, V(X) = np(1-p)Recall what n and p areWhat does σX equal ? Let X be the number of Americans with credit cards that pay off the full monthly balanceTake a sample size of 3 The probability of success is 0.59Find V(X) and σXContinuous Random VariableRecall that the expected value, E(X), for a continuous r.v. is similar to the case for a finite r.v.What are these formulas?The same similarity occurs for the formulas for the varianceIf X is a continuous random variable, what is V(X) and σX?ExampleLet X be the amount of time in minutes between arrivals/departures of planes at the Phoenix Sky Harbor airportWe know that α = 1.2What is fX(x)?Set up, but do not evaluate, an integral that corresponds to E(X)V(X)What happens when you integrate?Example #2Let X be a continuous random variable with a uniform distribution over the interval [0,700]What is fX(x)?What is µX?Using the above information, find V(X) σXSamplesWhen the distribution of a random variable X is unknown, the pmf or pdf can be approximated by the histogram of a random sample for X and the mean of X can be estimated by the sample meanHow did we denote the sample mean?Likewise for the variance of XHow do we denote the sample variance?How do we denote the sample standard deviation?ExamplesA random sample for Y is given by8,8,13,9,10,7,12,11,12,13Find s and s2A random sample for X, the number of Americans in a sample size 4 that have at least 1 credit card, is given by 2, 3, 4, 4, 3Find s and s2Summary ChartTaken directly from the course files, we have the following summary:RANDOM VARIABLE, XType Finite ContinuousValuesA finite set of numbersx1, x2, x3, , xnAll numbers in an intervalMeanVarianceSAMPLEValuesx1, x2, x3, , xnMeanVariance dxxfxXEXX)()(dxxfxXVXX)()()(2xall)()( xfxXEXXxall2)()()( xfxXVXX.11niixnx niixxns12211Sample Mean as a R.V.We have a statistic, , that is being used to estimate the expected value This statistic varies from sample to sampleI.e., it takes on different valuesThis statistic is a new random variableWe know the expected value of this random variable – what is it?The variance and standard deviation of the sample mean are denoted by___xnnXVxVXx__)()(__Focus on the ProjectHave continuous random variables V and R.What do they represent?We can now calculate the sample standard deviations of these random variablesSince we have no way of knowing the actual standard deviations of these random variables (why?), we will assume that the sample standard deviations are the same as the standard deviationsLet M be the continuous random variable that gives the sample mean for a set of 6 observations of R.Then, E(M) = E(R) and 6RMStandardizationSince we have two parameters, standard deviation and mean, we would like to know if knowing these two will characterize the distribution of a random variable, X.If so, we could use these to determine probabilitiesIt turns out that we cannotWe can have two different r.v’s, X and Y Have completely different shapes and different probabilitiesThey can have the same mean and standard deviationStandardizationCan simplify computations by transforming a random variableOne of the most common is standardizationThe standardization of X is given byIt can be shown that E(S) = 0 and σS = 1What does this mean?This idea of standardization will continue throughout the rest of this project
View Full Document