### Practice Problem Set 3 – basic lognormal problems

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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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$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

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