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Probability practice

Continuous PDFs and CDFs worksheets

Normalize bounded polynomial densities, integrate probabilities and moments, build piecewise CDFs, recover PDFs, and invert CDFs for quantiles.

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Show normalization or integral work. State complete piecewise functions with boundaries, and round only the final nonexact result to four decimals.

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Connect height, area, and accumulation

Every problem prints the bounded support and the zero-density assumption outside it. Learners normalize first, integrate areas and moments, and keep the complete boundary branches when moving between a PDF and CDF.

Learning objective

What this activity builds

Connect a valid continuous density to accumulated area, a full CDF, quantiles, zero point mass, expectation, and population variance.

Worked example

Let f(x)=2x on 0 <= x <= 1 and 0 otherwise. Find the full CDF.

F(x)=0 for x<=0; F(x)=x^2 for 0<x<1; F(x)=1 for x>=1.

Integrate the density from the lower support boundary to x, then add the constant outside-support branches.

Teaching notes

Make the practice count

  • Check nonnegativity and total area before using a proposed PDF.
  • Write all three CDF branches, including the support boundaries.
  • Distinguish density height from interval probability.

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Math SheetNameScoreTeacherDateContinuous PDFs and CDFsTreat a density value as height, not probability. Verify nonnegativity and total area 1, use definite integrals for interval probabilities and moments, state the full piecewise CDF includingoutside the support, and remember that a single point has probability 0. Give exact values where practical; otherwise round the final value to four decimals.1.Is f(x) = 3/2000 (x + 1)^2 on [-1, 9] and 0 otherwise a valid PDF? Justify both validity conditions.Given: X is measured in units. The displayed formula applies only on the bounded support [-1, 9]; the density is 0 elsewhere. X is continuous, so endpoints and individual pointshave probability 0.2.The CDF is F(x) = 0 for x <= -1; F(x) = (x + 1)^1/11 for -1 < x < 10; F(x) = 1 for x >= 10. Differentiate it to recover the full PDF.Given: X is measured in units. The displayed formula applies only on the bounded support [-1, 10]; the density is 0 elsewhere. X is continuous, so endpoints and individual pointshave probability 0.Density is height; probability is area. State support and boundaries.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateContinuous PDFs and CDFsTreat a density value as height, not probability. Verify nonnegativity and total area 1, use definite integrals for interval probabilities and moments, state the full piecewise CDF includingoutside the support, and remember that a single point has probability 0. Give exact values where practical; otherwise round the final value to four decimals.3.For the PDF f(x) = 1/8 (x - 2) on [2, 6], find E(X).Given: X is measured in units. The displayed formula applies only on the bounded support [2, 6]; the density is 0 elsewhere. X is continuous, so endpoints and individual pointshave probability 0.4.For f(x) = 1/10 on [-3, 7] and 0 otherwise, derive the full piecewise CDF F(x), including both outside-support branches.Given: X is measured in units. The displayed formula applies only on the bounded support [-3, 7]; the density is 0 elsewhere. X is continuous, so endpoints and individual pointshave probability 0.Density is height; probability is area. State support and boundaries.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateContinuous PDFs and CDFsTreat a density value as height, not probability. Verify nonnegativity and total area 1, use definite integrals for interval probabilities and moments, state the full piecewise CDF includingoutside the support, and remember that a single point has probability 0. Give exact values where practical; otherwise round the final value to four decimals.5.For f(x) = 2/121 (x - 3) on [3, 14] and 0 otherwise, find P(3 <= X <= 13).Given: X is measured in units. The displayed formula applies only on the bounded support [3, 14]; the density is 0 elsewhere. X is continuous, so endpoints and individual pointshave probability 0.6.On -2 <= x <= 8, let f(x) = k (x + 2)^2, and let f(x) = 0 otherwise. Find k so f is a valid PDF.Given: X is measured in units. The displayed formula applies only on the bounded support [-2, 8]; the density is 0 elsewhere. X is continuous, so endpoints and individual pointshave probability 0.Density is height; probability is area. State support and boundaries.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateContinuous PDFs and CDFsTreat a density value as height, not probability. Verify nonnegativity and total area 1, use definite integrals for interval probabilities and moments, state the full piecewise CDF includingoutside the support, and remember that a single point has probability 0. Give exact values where practical; otherwise round the final value to four decimals.7.For the continuous PDF f(x) = 3/64 (x + 1)^2 on [-1, 3], find P(X = 3).Given: X is measured in units. The displayed formula applies only on the bounded support [-1, 3]; the density is 0 elsewhere. X is continuous, so endpoints and individual pointshave probability 0.8.For the PDF f(x) = 1/2 on [0, 2], find Var(X).Given: X is measured in units. The displayed formula applies only on the bounded support [0, 2]; the density is 0 elsewhere. X is continuous, so endpoints and individual pointshave probability 0.Density is height; probability is area. State support and boundaries.math-sheet.comPage 4 of 4

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