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.

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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 2. Use the results in Exercise 2 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

  • 0.0968
  • 135.7267
  • 178.2442

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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 \gamma_1= 4.745329602
  • \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}

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