Practice Problem Set 8 – more lognormal calculation

Probability April 8, 2020 Comments: 2

This set of practice problems is to complement a discussion on lognormal distribution (found here).

Problems 8-G, 8-H and 8-I are on using the lognormal distribution as a model of security prices (see here).

Practice Problems

Practice Problem 8-A
Suppose the random variable X follows a lognormal distribution such that its 40th percentile is 19.9516 and its 60th percentile is 54.9649.

Determine the 80th percentile of X.

Practice Problem 8-B
Suppose the random variable X follows a lognormal distribution with parameters \mu=1 and \sigma=0.5. A random sample X_1,X_2,\cdots,X_{11} is drawn from a population represented by the random variable X. The associated order statistics are Y_1,Y_2,\cdots,Y_{11}.

The sample median is the 6th order statistic Y_6. Determine the probability P(Y_6<2.5).

Practice Problem 8-C
Same setting as in Problem 8-B. The 9th order statistic is the sample upper quartile. Determine the probability P(Y_9>4.5).

Practice Problem 8-D
Same setting as in Problem 8-B. Evaluate the following probabilities.

  • P(Y_6<2.5<Y_7<4.5<Y_9)
  • P(Y_6<2.5<4.5<Y_9)

Practice Problem 8-E
Same setting as in Problem 8-B. Evaluate the conditional probability P(Y_9>4.5 \lvert Y_6<2.5). Compare this with the unconditional probability P(Y_9>4.5). Does the answer for P(Y_9>4.5 \lvert Y_6<2.5) make sense in relation to P(Y_9>4.5)?

Practice Problem 8-F
Insurance claims follow a lognormal distribution with parameters \mu=3 and \sigma=2. Sixty four claims are currently processed by the insurer. Compute the following probabilities.

  • Find the probability that a randomly selected individual insurance claim whose amount between 100 and 250.
  • Find the probability that the average of the 64 claims is between 100 and 250.

Practice Problem 8-G
For a certain stock, the annual continuously compounded rate of return is modeled by a normal distribution with mean \mu=0.10 (10%) and \sigma=0.5. The initial amount of investment in this stock is 1000. At the end of one year, the value of the investment is modeled by the random variable Y=1000 \cdot e^{X}.

  • Determine the probability that the stock investment will increase in value at the end of one year.
  • Determine the probability that the stock investment will double in value or more at the end of one year.
  • Given that the stock investment increases in value at the end of one year, determine the probability that the investment will double in value or more.

Practice Problem 8-H
For a certain stock, the annual continuously compounded rate of return is modeled by a normal distribution with mean \mu=0.10 (10%) and \sigma=0.5. The initial amount of investment in this stock is 1000. Assume that the rates of return across the years are independent. At the end of t years, the value of the investment is modeled by the random variable

    Y=1000 \cdot e^{X_1+X_2+ \cdots + X_t}
  • Determine the probability that the stock investment will increase in value at the end of 5 years.
  • Determine the probability that the stock investment will double in value or more at the end of 5 years.
  • Given that the stock investment increases in value at the end of 5 years, determine the probability that the investment will double in value or more.

Practice Problem 8-I
Use the same setting as in Problem 8-H. Determine the probability that the stock investment increases in value over each of the next 5 years.

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    Answers

    Practice Problem 8-A

      \mu=3.5 and \sigma=2 after rounding

      80th percentile = e^{3.5+ 0.84 \times 2}=177.682811 (table value)

      80th percentile = e^{3.5+ 0.8416212335 \times 2}=178.2598767 (TI84+)

    Practice Problem 8-B

      P(Y_6<2.5)=0.3227058354 (table value)

      P(Y_6<2.5)=0.3252288245 (TI84+)

    Practice Problem 8-C

      P(Y_9>4.5)=0.2402227308 (table value)

      P(Y_9>4.5)=0.2417450602 (TI84+)

    Practice Problem 8-D

      P(Y_6<2.5<Y_7<4.5<Y_9)=0.0231966041 – two cases – table value
      P(Y_6<2.5<Y_7<4.5<Y_9)=0.0236006535 – two cases – TI84+

      P(Y_6<2.5<4.5<Y_9)=0.0303334 – six cases – table value
      P(Y_6<2.5<4.5<Y_9)=0.0312433215 – six cases – TI84+

    Practice Problem 8-E

      \displaystyle P(Y_9>4.5 \lvert Y_6<2.5)=\frac{P(Y_6<2.5<4.5<Y_9)}{P(Y_6<2.5)}

      \displaystyle \begin{aligned} P(Y_9>4.5 \lvert Y_6<2.5)&=\frac{P(Y_6<2.5<4.5<Y_9)}{P(Y_6<2.5)}\\& \\&=0.0939970607 \ \ (\text{table}) \\&=0.0960656596 \ \ (\text{TI84+}) \end{aligned}

    Practice Problem 8-F

      P(100<X<250)=0.10810 (table value)
      P(100<X<250)=0.1074041757 (TI84+)

      P(100<\overline{X}<250)=0.4140 (table value)
      P(100<\overline{X}<250)=0.4120092889 (TI84+)

    Practice Problem 8-G

      P(Y>1000)=0.5793 (table value)
      P(Y>1000)=0.5792596878 (TI84+)

      P(Y>2000)=0.1170 (table value)
      P(Y>2000)=0.1177530959 (TI84+)

      \displaystyle \begin{aligned} P(Y>2000 \lvert Y>1000)&=\frac{P(Y>2000)}{P(Y>1000)}\\& \\&=0.201967892 \ \ (\text{table}) \\&=0.2032820484 \ \ (\text{TI84+}) \end{aligned}

    Practice Problem 8-H

      P(Y>1000)=0.6736 (table value)
      P(Y>1000)=0.672639555 (TI84+)

      P(Y>2000)=0.4325 (table value)
      P(Y>2000)=0.4314215764 (TI84+)

      \displaystyle \begin{aligned} P(Y>2000 \lvert Y>1000)&=\frac{P(Y>2000)}{P(Y>1000)}\\& \\&=0.6420724466 \ \ (\text{table}) \\&=0.6413859743 \ \ (\text{TI84+}) \end{aligned}

    Practice Problem 8-H

      P(Y>1000)^5=P(X>0)^5=0.5793^5=0.0652405543 (table value)

      P(Y>1000)^5=P(X>0)^5=0.0652178578 (TI84+)

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2 thoughts on “Practice Problem Set 8 – more lognormal calculation”

  1. More lognormal calculation | Probability and Statistics Problem Solve April 8, 2020 at 2:25 pm Reply

    […] This post presents more calculation examples for lognormal distribution, complementing and supplementing previous posts on lognormal distribution. A practice problem set is found here. […]

  2. Lognormal model of security prices | Probability and Statistics Problem Solve April 8, 2020 at 9:15 pm Reply

    […] The preceding post discusses several examples of calculation involving the lognormal distribution. This post presents another one – using the lognormal distribution as a model of prices of a financial security. Practice problems are found here. […]

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