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Joint Continuous Distributions worksheets

Integrate densities on rectangular and triangular supports for probabilities, CDFs, marginals, conditionals, and independence.

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Every task prints the table, density support, paired observations, histogram bins, or sampling assumptions needed to reproduce the result.

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Use support geometry and iterated integrals.

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Verify support and total probability.

Support and normalization determine valid sums or integrals.

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  • Label support and units.
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Math SheetNameScoreTeacherDateJoint Continuous DistributionsUse the stated rectangular or triangular support. Integrate density over area; support restrictions are part of every marginal, conditional, and independence argument.1.The joint density is constant c on S={(x,y): 0<=x<=3.2, 0<=y<=12.7} and 0 elsewhere. Find E(X).Assumptions: The density is zero outside the explicitly stated support; all integrals use the shown coordinate units.Probability is a double integral of density over support. A point has probability 0; conditional and marginal densities include their support restrictions.2.The joint density is constant c on S={(x,y): x>=0, y>=0, x+y<=6.1} and 0 elsewhere. Find the full marginal densities fX and fY, including support.Assumptions: The density is zero outside the explicitly stated support; all integrals use the shown coordinate units.Probability is a double integral of density over support. A point has probability 0; conditional and marginal densities include their support restrictions.Show support, sums or integrals, and approximation assumptions.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateJoint Continuous DistributionsUse the stated rectangular or triangular support. Integrate density over area; support restrictions are part of every marginal, conditional, and independence argument.3.The joint density is constant c on S={(x,y): 0<=x<=4.4, 0<=y<=11.5} and 0 elsewhere. Find P(X<=2.2, Y<=5.75).Assumptions: The density is zero outside the explicitly stated support; all integrals use the shown coordinate units.Probability is a double integral of density over support. A point has probability 0; conditional and marginal densities include their support restrictions.4.The joint density is constant c on S={(x,y): 0<=x<=8.4, 0<=y<=7.5} and 0 elsewhere. Find c.Assumptions: The density is zero outside the explicitly stated support; all integrals use the shown coordinate units.Probability is a double integral of density over support. A point has probability 0; conditional and marginal densities include their support restrictions.Show support, sums or integrals, and approximation assumptions.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateJoint Continuous DistributionsUse the stated rectangular or triangular support. Integrate density over area; support restrictions are part of every marginal, conditional, and independence argument.5.The joint density is constant c on S={(x,y): 0<=x<=8.2, 0<=y<=7.7} and 0 elsewhere. Find F(4.1,3.85)=P(X<=4.1,Y<=3.85)Assumptions: The density is zero outside the explicitly stated support; all integrals use the shown coordinate units.Probability is a double integral of density over support. A point has probability 0; conditional and marginal densities include their support restrictions.6.The joint density is constant c on S={(x,y): 0<=x<=2.4, 0<=y<=13.5} and 0 elsewhere. Find the full conditional density f(Y | X=1.2), including valuesoutside conditional support.Assumptions: The density is zero outside the explicitly stated support; all integrals use the shown coordinate units.Probability is a double integral of density over support. A point has probability 0; conditional and marginal densities include their support restrictions.Show support, sums or integrals, and approximation assumptions.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateJoint Continuous DistributionsUse the stated rectangular or triangular support. Integrate density over area; support restrictions are part of every marginal, conditional, and independence argument.7.The joint density is constant c on S={(x,y): x>=0, y>=0, x+y<=12.1} and 0 elsewhere. Are X and Y independent? Use density and support reasoning.Assumptions: The density is zero outside the explicitly stated support; all integrals use the shown coordinate units.Probability is a double integral of density over support. A point has probability 0; conditional and marginal densities include their support restrictions.8.The joint density is constant c on S={(x,y): 0<=x<=13, 0<=y<=2.9} and 0 elsewhere. Find P(X<=6.5, Y<=1.45).Assumptions: The density is zero outside the explicitly stated support; all integrals use the shown coordinate units.Probability is a double integral of density over support. A point has probability 0; conditional and marginal densities include their support restrictions.Show support, sums or integrals, and approximation assumptions.math-sheet.comPage 4 of 4

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