Lognormal Distribution
Practice Problem Set 3 – basic lognormal problems
This post has several practice problems to go with this previous discussion on lognormal distribution.
Practice Problem 3A 
The amount of annual losses from an insured follows a lognormal distribution with parameters and = 0.6 and with mode = 2.5. Calculate the mean annual loss for a randomly selected insured. 
Practice Problem 3B 
Claim size for an auto insurance coverage follows a lognormal distribution with mean 149.157 and variance 223.5945. Determine the probability that a randomly selected claim will be greater than 170. 
Practice Problem 3C 
For xray machines produced by a certain manufacturer, the following is known.
Calculate the probability that an xray machine produced by this manufacturer will last at least 12 years. 
Practice Problem 3D 
Claim sizes expressed in US dollars follow a lognormal distribution with parameters = 5 and = 0.25. One Canadian dollar is currently worth $0.75 US dollars. Calculate the 75th percentile of a claim in Canadian dollars. 
Practice Problem 3E 
For a commercial fire coverage, the size of a loss follows a lognormal distribution with parameters = 2.75 and = 0.75. Determine where is the 75th percentile of a loss and is the 25th percentile of a loss. Note that is known as the interquartile range. 
Practice Problem 3F 
Claim sizes in the current year follow a lognormal distribution with = 4.75 and = 0.25. In the next year, all claims are expected to be inflated uniformly by 25%. One claim is expected in the next year for an insured. Determine where is the 80th percentile of the size of this claim and is the 40th percentile of the size of this claim. 
Practice Problem 3G 
Determine the median of the portion of next year’s loss distribution that is above 10. 
Practice Problem 3H 
Losses follow a lognormal distribution with mean 17 and variance 219. Determine the skewness of the loss distribution. 
Practice Problem 3I 
Determine the probability that a loss is less than 5000. 
Practice Problem 3J 
Determine the mean of the losses. 
All normal probabilities are obtained by using the normal distribution table found here.
Problem  Answer 

3A  4.29 
3B  0.0869 
3C  0.2033 
3D  233.9675 
3E  16.39085 
3F  42.5155 
3G  21.143268 
3H  3.271185 
3I  0.7764 
3J  1124.394559 
______________________________________________________________________________________________________________________________
2017 – Dan Ma
Basic properties of lognormal distribution
A detailed discussion of the mathematical properties of lognormal distribution is found in this previous post in a companion blog. This post shows how to work basic calculation problems for lognormal distribution. A summary of lognormal distribution is given and is followed by several examples. Practice problems are in the next post.
______________________________________________________________________________________________________________________________
Basic Properties
The random variable is said to follow a lognormal distribution with parameters and if follows a normal distribution with mean and variance . Here, is the natural logarithm in base = 2.718281828…. It is difficult (if not impossible) to calculate probabilities by integrating the lognormal density function. Since the lognormal distribution is intimately related to the normal distribution, the basic lognormal calculation is performed by calculating the corresponding normal distribution. The following summary shows how.
In the following points, has a lognormal distribution with parameters and and is the corresponding normal distribution with mean and variance . The notation means raising to the number .
1. Lognormal observations and normal observations 

2. Lognormal CDF and normal CDF 

3. Lognormal density function and normal density function 

4. Lognormal moments and normal moment generating function 

5. Examples of lognormal moments 

6. Lognormal percentiles and normal percentiles 

7. Constant multiple of lognormal distribution 

8. Mode of lognormal distribution 

______________________________________________________________________________________________________________________________
Examples
Two examples are given to illustrate the calculation discussed here. The next post has practice problems.
All normal probabilities are obtained by using the normal distribution table found here.
Example 1
Suppose that the random variable has a lognormal distribution with parameters = 1 and = 2. Calculate the following.
 and
 The 67th, 95th and 99th percentiles of .
 Let . Find and
_______________________________________________
To find the percentiles, first find the standard normal percentiles, either by using calculator or by looking up a table. Using a standard normal table, we get 0.44 (67th percentile), 1.645 (95th percentile) and 2.33 (99th percentile). The following gives the lognormal percentiles.

(67th percentile)
(95th percentile)
(99th percentile)
_______________________________________________
The random variable has a lognormal distribution with parameters and = 2.
Note. One interpretation of is that of inflation, in this case a 10% inflation. For example, let be the size of a randomly selected auto insurance collision claim in the current year. If the claims are expected to increase 10% in the following year, is the the size of a randomly selected claim in the following year.
Example 2
Suppose that the random variable has a lognormal distribution with mean 12.18 and variance 255.02. Calculation the following.
 The skewness and kurtosis of .
First, determine the parameters and by setting up the following equations.
Plug the first equation into the second equation and obtain the equation . Solving for produces = 1. Plug = 1 into the first equation produces = 2. The following gives the desired probability.
To find the skewness and kurtosis, one way is to find the first 4 lognormal moments and then calculate the third standardized moment (skewness) and the fourth standardized moment (kurtosis). To see how this is done, see this previous post. Another is to use the formulas given above.
Example 3
Suppose that the lifetime (in years) of a certain type of machines follows the lognormal distribution described in Example 2. Suppose that you purchased such a machine that is 10year old. What is the probability that it will last another 10 years?
This is a conditional probability since the machine already survived for 10 years already.
______________________________________________________________________________________________________________________________
2017 – Dan Ma