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Continuous Random Variable Transformations worksheets

Transform supports, CDFs, and PDFs for affine, square, square-root, and two-preimage mappings.

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Write the transformed support, full piecewise CDF, and PDF. Account for decreasing maps and every valid preimage.

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Learning objective

What this activity builds

Derive a transformed continuous distribution with correct support, boundary pieces, monotonic direction, and Jacobian contributions.

Worked example

X is uniform on 0 to 2 and Y=X squared. Find F_Y(y) inside the support.

F_Y(y)=sqrt(y)/2 for 0<=y<=4.

On the nonnegative support, X squared is increasing.

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  • Transform the endpoints first.
  • Reverse inequalities for decreasing maps.
  • Count both preimages for a symmetric square.

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Math SheetNameScoreTeacherDateContinuous Random Variable TransformationsWrite the transformed support, full piecewise CDF, and PDF. Account for decreasing maps and every valid preimage.1.X is Uniform(0,17) and Y=X^2. Write the support and full piecewise CDF and PDF of Y.Reference: On nonnegative support, x maps monotonically to x^2 and the inverse is sqrt(y).2.X is Uniform(0,18) and Y=3X+0. Write the support and full piecewise CDF and PDF of Y.Reference: For a monotone map, solve the inequality for X. A negative slope reverses it; the density divides by the absolute derivative.State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateContinuous Random Variable TransformationsWrite the transformed support, full piecewise CDF, and PDF. Account for decreasing maps and every valid preimage.3.X is Uniform(0,2) and Y=-4X+1. Write the support and full piecewise CDF and PDF of Y.Reference: For a monotone map, solve the inequality for X. A negative slope reverses it; the density divides by the absolute derivative.4.X is Uniform(-3,3) and Y=X^2. Write the support and full piecewise CDF and PDF of Y, accounting for both preimages.Reference: For 0<y<upper^2, the event X^2<=y is -sqrt(y)<=X<=sqrt(y); both inverse branches contribute to the density.State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateContinuous Random Variable TransformationsWrite the transformed support, full piecewise CDF, and PDF. Account for decreasing maps and every valid preimage.5.X is Uniform(0,16) and Y=sqrt(X). Write the support and full piecewise CDF and PDF of Y.Reference: The inverse is x=y^2. Differentiate F_X(y^2) on the transformed support.6.X is Uniform(0,25) and Y=sqrt(X). Write the support and full piecewise CDF and PDF of Y.Reference: The inverse is x=y^2. Differentiate F_X(y^2) on the transformed support.State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateContinuous Random Variable TransformationsWrite the transformed support, full piecewise CDF, and PDF. Account for decreasing maps and every valid preimage.7.X is Uniform(-6,6) and Y=X^2. Write the support and full piecewise CDF and PDF of Y, accounting for both preimages.Reference: For 0<y<upper^2, the event X^2<=y is -sqrt(y)<=X<=sqrt(y); both inverse branches contribute to the density.8.X is Uniform(0,7) and Y=-3X+1. Write the support and full piecewise CDF and PDF of Y.Reference: For a monotone map, solve the inequality for X. A negative slope reverses it; the density divides by the absolute derivative.State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 4 of 4

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