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Covariance of Random Variables worksheets

Calculate covariance and correlation from joint PMFs or paired data with explicit denominators.

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Relate joint moments, covariance, correlation, and linear-combination variance.

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Math SheetNameScoreTeacherDateCovariance of Random VariablesState whether data are a population or sample. Use the specified denominator, keep paired observations together, and distinguish zero covariance from independence.1.Let X be equally likely to be -11.3, 0, or 11.3, and define Y=X^2. Show Cov(X,Y)=0 and decide whether X and Y are independent.Assumptions: The stated observations and denominator convention define the dataset; no pairs are omitted.Paired products use matching observations. Population covariance divides by n; sample covariance divides by n-1.2.A joint PMF assigns probability 1/6 to each pair (63,128), (69,144), (69,148), (75,152), (75,156), (81,172). Find Cov(X,Y) using E(XY)-E(X)E(Y).Assumptions: The stated observations and denominator convention define the dataset; no pairs are omitted.Paired products use matching observations. Population covariance divides by n; sample covariance divides by n-1.Show support, sums or integrals, and approximation assumptions.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateCovariance of Random VariablesState whether data are a population or sample. Use the specified denominator, keep paired observations together, and distinguish zero covariance from independence.3.Paired observations (7 pairs): (56,60), (60,66), (58,66), (62,66), (60,66), (64,72), (62,66). Treat these as a sample. Find covariance using denominatorn-1.Assumptions: The stated observations and denominator convention define the dataset; no pairs are omitted.Paired products use matching observations. Population covariance divides by n; sample covariance divides by n-1.4.Paired observations (5 pairs): (-74,-222), (-70,-208), (-70,-206), (-66,-198), (-66,-196). Treat these as the whole population. Find covariance usingdenominator n.Assumptions: The stated observations and denominator convention define the dataset; no pairs are omitted.Paired products use matching observations. Population covariance divides by n; sample covariance divides by n-1.Show support, sums or integrals, and approximation assumptions.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateCovariance of Random VariablesState whether data are a population or sample. Use the specified denominator, keep paired observations together, and distinguish zero covariance from independence.5.Paired observations (5 pairs): (70,215), (73,225), (74,229), (77,236), (78,240). Treat these as the whole population. Find Var(2X-Y).Assumptions: The stated observations and denominator convention define the dataset; no pairs are omitted.Paired products use matching observations. Population covariance divides by n; sample covariance divides by n-1.6.Paired observations (8 pairs): (49,197), (53,218), (53,223), (57,229), (57,234), (61,255), (61,245), (65,266). Treat these as the whole population. FindCov(3X+2,3Y+4).Assumptions: The stated observations and denominator convention define the dataset; no pairs are omitted.Paired products use matching observations. Population covariance divides by n; sample covariance divides by n-1.Show support, sums or integrals, and approximation assumptions.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateCovariance of Random VariablesState whether data are a population or sample. Use the specified denominator, keep paired observations together, and distinguish zero covariance from independence.7.Paired observations (5 pairs): (22,19), (27,27), (26,29), (31,28), (30,30). Treat these as the whole population. Find correlation.Assumptions: The stated observations and denominator convention define the dataset; no pairs are omitted.Paired products use matching observations. Population covariance divides by n; sample covariance divides by n-1.8.Paired observations (6 pairs): (-17,-52), (-13,-36), (-15,-38), (-11,-34), (-13,-36), (-9,-20). Treat these as the whole population. Find covariance usingdenominator n.Assumptions: The stated observations and denominator convention define the dataset; no pairs are omitted.Paired products use matching observations. Population covariance divides by n; sample covariance divides by n-1.Show support, sums or integrals, and approximation assumptions.math-sheet.comPage 4 of 4

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