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 \mu and \sigma = 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 x-ray machines produced by a certain manufacturer, the following is known.

  • Lifetime in years follows a lognormal distribution with \mu = 0.9 and \sigma.
  • The expected lifetime of such machines is 15 years.

Calculate the probability that an x-ray 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 \mu = 5 and \sigma = 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 \mu = 2.75 and \sigma = 0.75. Determine y-x where y is the 75th percentile of a loss and x is the 25th percentile of a loss. Note that y-x is known as the interquartile range.

Practice Problem 3F

Claim sizes in the current year follow a lognormal distribution with \mu = 4.75 and \sigma = 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 y-x where y is the 80th percentile of the size of this claim and x is the 40th percentile of the size of this claim.

Practice Problem 3G
  • In the current year, losses follow a lognormal distribution with \mu = 1.6 and \sigma = 1.35.
  • In the next year, inflation of 20% will impact all losses uniformly.
  • 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
  • Losses from an insurance coverage follow a lognormal distribution with parameters \mu and \sigma = 2.
  • The 80th percentile of the losses is 5884.
  • Determine the probability that a loss is less than 5000.

Practice Problem 3J
  • Losses from an insurance coverage follow a lognormal distribution.
  • The 25th percentile of the losses is 133.62.
  • The 75th percentile of the losses is 997.25.
  • Determine the mean of the losses.

All normal probabilities are obtained by using the normal distribution table found here.

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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

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\copyright 2017 – Dan Ma

3 thoughts on “Practice Problem Set 3 – basic lognormal problems

  1. Pingback: Lognormal distribution | Topics in Actuarial Modeling

  2. Pingback: Basic exercises for lognormal distribution | Probability and Statistics Problem Solve

  3. Pingback: Basic properties of lognormal distribution « Practice Problems in Actuarial Modeling

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