Monthly Archives: October 2015

Basic exercises for lognormal distribution

This post presents exercises on the lognormal distribution. These exercises are to reinforce the basic properties discussed in this companion blog post.

Additional resources: another discussion of lognormal, a concise summary of lognormal and a problem set on lognormal.

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Exercises

Exercise 1
Let X be a normal random variable with mean 6.5 and standard deviation 0.8. Consider the random variable Y=e^X. what is the probability P(800 \le Y \le 1000)?

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Exercise 2
Suppose Y follows a lognormal distribution with parameters \mu=1 and \sigma=1. Let Y_1=1.25 Y. Determine the following:

  • The probability that Y_1 exceed 1.
  • The 40th percentile of Y_1.
  • The 80th percentile of Y_1.

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Exercise 3
Let Y follows a lognormal distribution with parameters \mu=4 and \sigma=0.9. Compute the mean, second moment, variance, third moment and the fourth moment.

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Exercise 4
Let Y be the same lognormal distribution as in Exercise 3. Use the results in Exercise 3 to compute the coefficient of variation, coefficient of skewness and the kurtosis.

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Exercise 5
Given the following facts about a lognormal distribution:

  • The lower quartile (i.e. 25% percentile) is 1000.
  • The upper quartile (i.e. 75% percentile) is 4000.

Determine the mean and variance of the given lognormal distribution.

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Exercise 6
Suppose that a random variable Y follows a lognormal distribution with mean 149.157 and variance 223.5945. Determine the probability P(Y>150).

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Exercise 7
Suppose that a random variable Y follows a lognormal distribution with mean 1200 and median 1000. Determine the probability P(Y>1300).

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Exercise 8
Customers of a very popular restaurant usually have to wait in line for a table. Suppose that the wait time Y (in minutes) for a table follows a lognormal distribution with parameters \mu=3.5 and \sigma=0.10. Concerned about long wait time, the restaurant owner improves the wait time by expanding the facility and hiring more staff. As a result, the wait time for a table is cut by half. After the restaurant expansion,

  • what is the probability distribution of the wait time for a table?
  • what is the probability that a customer will have to wait more than 20 minutes for a table?

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Answers

Exercise 1

  • 0.1040

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Exercise 2

  • 0.8888
  • 1.4669
  • 7.8707

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Exercise 3

  • E(Y)=e^{4.405}
  • E(Y^2)=e^{9.62}
  • E(Y^3)=e^{15.645}
  • E(Y^4)=e^{22.48}
  • Var(Y)=(e^{0.81}-1) \ e^{8.81}

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Exercise 4

  • \displaystyle \text{CV}= 1.117098
  • \displaystyle \gamma_1= 4.74533
  • \displaystyle \beta_2= 60.41075686

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Exercise 5

  • \displaystyle E(Y)= 3415.391045
  • \displaystyle E(Y^2)= 34017449.61
  • \displaystyle Var(Y)= 22352553.62

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Exercise 6

  • 0.4562

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Exercise 7

  • 0.3336

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Exercise 8

  • longnormal with \mu=3.5+\log(0.5) and \sigma=0.1 where \log is the natural logarithm.
  • 0.0294

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\copyright \ 2015 \text{ by Dan Ma}
Revised October 18, 2018

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