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Exponential Distribution worksheets

Work with exponential waiting-time CDFs, survival probabilities, intervals, quantiles, moments, rate-mean conversion, and memorylessness.

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Let T be a nonnegative waiting time with positive rate lambda. Convert rate and mean in matching units. Use a calculator for exponential and logarithm values. Round final probabilities to the configured 3 or 4 decimal places; use scientific notation when a rounded display would be 0. Round requested quantiles as stated.

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Keep the rate and units visible

Poisson sheets distinguish a rate per unit from lambda for the full exposure. Exponential sheets label lambda as a positive rate and connect it to the reciprocal mean waiting time.

Learning objective

What this activity builds

Interpret a positive exponential rate, convert between rate and mean, and choose the correct waiting-time probability, quantile, or memoryless relation.

Worked example

If lambda=1/2 per minute, find P(T>4).

P(T>4)=exp[-(1/2)(4)]=exp(-2), approximately 0.135.

The survival function gives the probability that the wait exceeds the stated time.

Teaching notes

Make the practice count

  • Check that rate and time use reciprocal units.
  • Use survival values directly for upper tails and interval differences.
  • For memorylessness, separate elapsed time s from additional time t.

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Math SheetNameScoreTeacherDateExponential DistributionLet T be a nonnegative waiting time with positive rate lambda. Convert between rate and mean in matching units. Use a calculator for exponential and logarithm values. Round finalprobabilities to the 3 decimal places; if that would display 0, use scientific notation. Round requested quantiles as stated.1.Let T be an exponential waiting time in hours with mean waiting time 0.8 hours, so lambda = 1/mean > 0. Find the mean and population variance of T.Assumptions: The waiting time is nonnegative and exponential with one constant positive rate; the process is stationary and memoryless.Reference: F(t)=1-exp(-lambda t), P(T>t)=exp(-lambda t), and mean=1/lambda for t>=0. Calculator required for exponential and logarithm values.2.Let T be an exponential waiting time in minutes with positive rate lambda = 0.5 per minute. Find P(2 < T <= 10).Assumptions: The waiting time is nonnegative and exponential with one constant positive rate; the process is stationary and memoryless.Reference: F(t)=1-exp(-lambda t), P(T>t)=exp(-lambda t), and mean=1/lambda for t>=0. Calculator required for exponential and logarithm values.Keep rate and exposure units consistent. Round only the final result.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateExponential DistributionLet T be a nonnegative waiting time with positive rate lambda. Convert between rate and mean in matching units. Use a calculator for exponential and logarithm values. Round finalprobabilities to the 3 decimal places; if that would display 0, use scientific notation. Round requested quantiles as stated.3.Let T be an exponential waiting time in hours with mean waiting time 5 hours, so lambda = 1/mean > 0. Find P(T > 8).Assumptions: The waiting time is nonnegative and exponential with one constant positive rate; the process is stationary and memoryless.Reference: F(t)=1-exp(-lambda t), P(T>t)=exp(-lambda t), and mean=1/lambda for t>=0. Calculator required for exponential and logarithm values.4.Let T be an exponential waiting time in hours with positive rate lambda = 0.125 per hour. Find q such that P(T <= q) = 0.2. Round q to the nearestthousandth.Assumptions: The waiting time is nonnegative and exponential with one constant positive rate; the process is stationary and memoryless.Reference: F(t)=1-exp(-lambda t), P(T>t)=exp(-lambda t), and mean=1/lambda for t>=0. Calculator required for exponential and logarithm values.Keep rate and exposure units consistent. Round only the final result.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateExponential DistributionLet T be a nonnegative waiting time with positive rate lambda. Convert between rate and mean in matching units. Use a calculator for exponential and logarithm values. Round finalprobabilities to the 3 decimal places; if that would display 0, use scientific notation. Round requested quantiles as stated.5.Let T be an exponential waiting time in minutes with positive rate lambda = 0.25 per minute. Given that T > 0.5, find P(T > 6.5 | T > 0.5).Assumptions: The waiting time is nonnegative and exponential with one constant positive rate; the process is stationary and memoryless.Reference: F(t)=1-exp(-lambda t), P(T>t)=exp(-lambda t), and mean=1/lambda for t>=0. Calculator required for exponential and logarithm values.6.Let T be an exponential waiting time in minutes with mean waiting time 0.5 minutes, so lambda = 1/mean > 0. Find P(T <= 4).Assumptions: The waiting time is nonnegative and exponential with one constant positive rate; the process is stationary and memoryless.Reference: F(t)=1-exp(-lambda t), P(T>t)=exp(-lambda t), and mean=1/lambda for t>=0. Calculator required for exponential and logarithm values.Keep rate and exposure units consistent. Round only the final result.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateExponential DistributionLet T be a nonnegative waiting time with positive rate lambda. Convert between rate and mean in matching units. Use a calculator for exponential and logarithm values. Round finalprobabilities to the 3 decimal places; if that would display 0, use scientific notation. Round requested quantiles as stated.7.Let T be an exponential waiting time in hours with mean waiting time 2 hours, so lambda = 1/mean > 0. Find P(T > 4).Assumptions: The waiting time is nonnegative and exponential with one constant positive rate; the process is stationary and memoryless.Reference: F(t)=1-exp(-lambda t), P(T>t)=exp(-lambda t), and mean=1/lambda for t>=0. Calculator required for exponential and logarithm values.8.Let T be an exponential waiting time in hours with positive rate lambda = 0.2 per hour. Given that T > 2, find P(T > 7 | T > 2).Assumptions: The waiting time is nonnegative and exponential with one constant positive rate; the process is stationary and memoryless.Reference: F(t)=1-exp(-lambda t), P(T>t)=exp(-lambda t), and mean=1/lambda for t>=0. Calculator required for exponential and logarithm values.Keep rate and exposure units consistent. Round only the final result.math-sheet.comPage 4 of 4

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