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Discrete CDFs, Transformations, and Moments worksheets

Build step CDFs, find quantiles, merge transformed PMFs, and apply weighted moment rules.

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Use F(x)=P(X<=x). For quantiles, choose the smallest support value whose CDF reaches the target.

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Learning objective

What this activity builds

Move accurately among finite PMFs, CDFs, quantiles, transformations, expectation, variance, and standard deviation.

Worked example

If X is equally likely to be -1 or 1, find the PMF of X squared.

P(X squared=1)=1.

Both original support points merge at one transformed value.

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Make the practice count

  • Accumulate CDF probabilities from left to right.
  • Merge equal transformed values.
  • Square the affine scale factor for variance.

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Math SheetNameScoreTeacherDateDiscrete CDFs, Transformations, and MomentsUse the convention F(x)=P(X<=x). For quantiles, choose the smallest support value whose CDF reaches the target.1.X has support -3, -1, 1, 3 with probabilities 0.2, 0.3, 0.1, 0.4. For Y=X^2, give the PMF and merge repeated transformed values.Reference: When several X values map to one Y value, add their probabilities.2.X has mean 10 and variance 7. Let Y=-3X+3. Find E(Y), Var(Y), and SD(Y).Reference: E(aX+b)=aE(X)+b; Var(aX+b)=a^2 Var(X); SD scales by |a|.State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateDiscrete CDFs, Transformations, and MomentsUse the convention F(x)=P(X<=x). For quantiles, choose the smallest support value whose CDF reaches the target.3.X is equally likely to be 6 or 10; Y is equally likely to be 4 or 12. Compare their means and standard deviations.Reference: Equal means do not imply equal spread. Compute probability-weighted variance around the common mean.4.For support 7, 9, 11, 13 with probabilities 0.25, 0.25, 0.25, 0.25, evaluate F(6), F(10), and F(14).Reference: A discrete CDF is right-continuous and constant between support points.State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateDiscrete CDFs, Transformations, and MomentsUse the convention F(x)=P(X<=x). For quantiles, choose the smallest support value whose CDF reaches the target.5.X has support 8, 10, 12, 14, 16 with probabilities 0.1, 0.15, 0.2, 0.25, 0.3 in matching order. Make a PMF table and a step-CDF table.Reference: F(x)=P(X<=x). The CDF is 0 below the support and 1 at or above its largest value.6.Independent X and Y have E(X)=11.6, Var(X)=2.44, E(Y)=2, and Var(Y)=2. Find the mean, variance, and SD of X+Y.Reference: Means always add. Variances add here because independence makes covariance zero.State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateDiscrete CDFs, Transformations, and MomentsUse the convention F(x)=P(X<=x). For quantiles, choose the smallest support value whose CDF reaches the target.7.X has support 10, 12, 14, 16 with probabilities 0.25, 0.25, 0.25, 0.25. Find E(X), E(X^2), Var(X), and SD(X).Reference: Use probability-weighted sums; Var(X)=E(X^2)-[E(X)]^2 and SD is the positive square root.8.X has support 11, 13, 15, 17, 19 with probabilities 0.1, 0.15, 0.2, 0.25, 0.3. Using q_p=min{x in support: F(x)>=p}, find q_0.7.Reference: Choose the first support point whose cumulative probability reaches or exceeds p.State the model or rule, show exact setup, then round only the final decimal.math-sheet.comPage 4 of 4

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