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Uniform Distribution worksheets

Use support length for density, clipped interval and conditional probabilities, piecewise CDFs, quantiles, mean, and variance.

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Use density 1/(b-a), clip intervals to the support, show conditional overlap ratios, and state squared units for variance.

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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

Model a continuous uniform variable on [a,b] and reason correctly about boundaries, outside-support intervals, conditional overlap, quantiles, and moments.

Worked example

If X is uniform on [2,10], find P(0 <= X <= 6).

The overlap with the support is [2,6], so P=4/8=1/2.

Only overlap length inside the support contributes probability.

Teaching notes

Make the practice count

  • Draw the support before measuring interval overlap.
  • For conditional intervals, divide the joint overlap length by the conditioning overlap length.
  • Keep variance in squared units.

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Math SheetNameScoreTeacherDateUniform DistributionFor X uniform on the stated closed interval [a,b], use density 1/(b-a) on the support and 0 outside it. Clip probability intervals to the support, divide overlap lengths for conditionalintervals, include the boundary branches of the CDF, and give exact values where practical; otherwise round the final value to four decimals.1.Let X be uniform on [-3, -1]. Find the 40th percentile q, defined by P(X <= q) = 0.4.Given: X is measured in units and is continuous with constant density on [-3, -1] and density 0 outside; endpoints have probability 0.2.Let X be uniform on [-6, 5]. Find Var(X). State the squared units.Given: X is measured in units and is continuous with constant density on [-6, 5] and density 0 outside; endpoints have probability 0.Density is height; probability is area. State support and boundaries.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateUniform DistributionFor X uniform on the stated closed interval [a,b], use density 1/(b-a) on the support and 0 outside it. Clip probability intervals to the support, divide overlap lengths for conditionalintervals, include the boundary branches of the CDF, and give exact values where practical; otherwise round the final value to four decimals.3.Let X be uniform on [-6, 0]. Find P(-6 <= X <= 0 | -8 <= X <= -1).Given: X is measured in units and is continuous with constant density on [-6, 0] and density 0 outside; endpoints have probability 0.4.Let X be uniform on [-4, 6]. Find P(X >= 5).Given: X is measured in units and is continuous with constant density on [-4, 6] and density 0 outside; endpoints have probability 0.Density is height; probability is area. State support and boundaries.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateUniform DistributionFor X uniform on the stated closed interval [a,b], use density 1/(b-a) on the support and 0 outside it. Clip probability intervals to the support, divide overlap lengths for conditionalintervals, include the boundary branches of the CDF, and give exact values where practical; otherwise round the final value to four decimals.5.Let X be uniform on [-5, -2]. Two model descriptions are proposed. Model A has constant PDF 1/3 on [-5, -2] and 0 outside. Model B assignsprobability d/3 to every subinterval of length d inside [-5, -2]. Select which descriptions represent Uniform(-5, -2), then state the support, fullpiecewise PDF, and full piecewise CDF.Given: X is measured in units and is continuous with constant density on [-5, -2] and density 0 outside; endpoints have probability 0.6.Let X be uniform on [5, 15]. Find P(3 <= X <= 17).Given: X is measured in units and is continuous with constant density on [5, 15] and density 0 outside; endpoints have probability 0.Density is height; probability is area. State support and boundaries.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateUniform DistributionFor X uniform on the stated closed interval [a,b], use density 1/(b-a) on the support and 0 outside it. Clip probability intervals to the support, divide overlap lengths for conditionalintervals, include the boundary branches of the CDF, and give exact values where practical; otherwise round the final value to four decimals.7.Let X be uniform on [-2, 1]. Find E(X).Given: X is measured in units and is continuous with constant density on [-2, 1] and density 0 outside; endpoints have probability 0.8.Let X be uniform on [-2, 2]. Let X be uniform on [-2, 2]. Write the full piecewise CDF, with correct boundary branches.Given: X is measured in units and is continuous with constant density on [-2, 2] and density 0 outside; endpoints have probability 0.Density is height; probability is area. State support and boundaries.math-sheet.comPage 4 of 4

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