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 be a normal random variable with mean 6.5 and standard deviation 0.8. Consider the random variable . what is the probability ?

*Exercise 2*

Suppose follows a lognormal distribution with parameters and . Let . Determine the following:

- The probability that exceed 1.
- The 40th percentile of .
- The 80th percentile of .

*Exercise 3*

Let follows a lognormal distribution with parameters and . Compute the mean, second moment, variance, third moment and the fourth moment.

*Exercise 4*

Let 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.

*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.

*Exercise 6*

Suppose that a random variable follows a lognormal distribution with mean 149.157 and variance 223.5945. Determine the probability .

*Exercise 7*

Suppose that a random variable follows a lognormal distribution with mean 1200 and median 1000. Determine the probability .

*Exercise 8*

Customers of a very popular restaurant usually have to wait in line for a table. Suppose that the wait time (in minutes) for a table follows a lognormal distribution with parameters and . 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

*Exercise 2*

- 0.8888
- 1.4669
- 7.8707

*Exercise 3*

*Exercise 4*

- 1.117098
- 4.74533
- 60.41075686

*Exercise 5*

- 3415.391045
- 34017449.61
- 22352553.62

*Exercise 6*

- 0.4562

*Exercise 7*

- 0.3336

*Exercise 8*

- longnormal with and where is the natural logarithm.
- 0.0294

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Revised October 18, 2018