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Magazine Chapter 4: Problem 1
Suppose that \(X\) has a lognormal distribution with parameters \(\theta=5\) and \(\omega^{2}=9 .\) Determine the following: a. \(P(X<13,300)\) b. Value for \(x\) such that \(P(X \leq x)=0.95\) c. Mean and variance of \(X\)
Short Answer
Expert verified
a. \(0.9332\), b. \(20,717.64\), c. Mean: \(13,396.9\), Variance: \(1,978,544,288.2\).
Step by step solution
01
Understanding Lognormal Distribution Parameters
In a lognormal distribution, if \(X\) is lognormally distributed with parameters \(\theta=\mu\) and \(\omega^{2}=\sigma^2\), then the natural logarithm of \(X\), \(\ln(X)\), follows a normal distribution with mean \(\mu\) and variance \(\sigma^2\). Here, \(\mu=5\) and \(\sigma^2=9\).
02
Convert Problem a to Normal Distribution
To find \(P(X<13,300)\), we first convert \(X\) to a standard normal variable. Calculate \(Z\) using the formula: \[Z = \frac{\ln(X) - \mu}{\sigma}\]Substitute \(X=13,300\), \(\mu=5\), and \(\sigma=3\) (since \(\sigma^2=9\)): \[Z = \frac{\ln(13,300) - 5}{3}\]
03
Calculate Z-Value for Part a
Calculate \(\ln(13,300)\) and substitute into the Z formula: \[\ln(13,300) \approx 9.5\]\[Z = \frac{9.5 - 5}{3} = \frac{4.5}{3} = 1.5\]Now find \(P(Z < 1.5)\) using the standard normal distribution tables.
04
Find Probability for Part a
Using the standard normal distribution table, \(P(Z < 1.5)\) is approximately 0.9332. Thus, \(P(X < 13,300) = 0.9332\).
05
Determine Z-Value for Part b
For part (b), we need to find \(x\) such that \(P(X \leq x) = 0.95\). Hence, find \(Z\) such that \(P(Z < z) = 0.95\). From the standard normal distribution table, \(Z \approx 1.645\)
06
Calculate X for Part b
Convert \(Z\) back to \(X\) using the formula: \[x = e^{\mu + \sigma \cdot Z}\]Substitute \(\mu=5\), \(\sigma=3\) and \(Z=1.645\): \[x = e^{5 + 3 \times 1.645} \approx e^{9.935} \approx 20,717.64\]
07
Calculate Mean for Part c
The mean of a lognormal distribution \(X\) is given by the formula: \[E(X) = e^{\mu + \frac{\sigma^2}{2}}\]Substitute \(\mu=5\) and \(\sigma^2=9\): \[E(X) = e^{5 + \frac{9}{2}} = e^{9.5} \approx 13,396.9\]
08
Calculate Variance for Part c
The variance of a lognormal distribution \(X\) is given by: \[\text{Var}(X) = \left(e^{\sigma^2} - 1\right) e^{2\mu + \sigma^2}\]Substitute \(\mu=5\) and \(\sigma^2=9\): \[\text{Var}(X) = \left(e^{9} - 1\right) e^{2(5) + 9} \approx 1,978,544,288.2\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Normal Distribution
The normal distribution is a foundational concept in statistics, often known as the bell curve due to its shape. A normal distribution is symmetric around its mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. The two key parameters of a normal distribution are the mean (\[\mu\]) and the variance (\[\sigma^{2}\]).
- **Mean** - Determines the center of the distribution.
- **Variance** - Reflects the spread of the distribution; larger values indicate that the data is spread out more.Most values cluster around a central region, with decreasing frequency as we move further away. This distribution is significant since many phenomena in the real world follow a normal distribution, such as heights, exam scores, or measurement errors.
- **Mean** - Determines the center of the distribution.
- **Variance** - Reflects the spread of the distribution; larger values indicate that the data is spread out more.Most values cluster around a central region, with decreasing frequency as we move further away. This distribution is significant since many phenomena in the real world follow a normal distribution, such as heights, exam scores, or measurement errors.
Standard Normal Variable
A standard normal variable is a special case of a normal distribution with a mean of zero and a standard deviation of one. This is typically denoted by the variable \[Z\]. The transformation to a standard normal variable allows us to compare scores from different normal distributions.
By using the formula:- \[Z = \frac{{X - \mu}}{{\sigma}}\]We convert a normal variable \[X\] (with mean \[\mu\] and standard deviation \[\sigma\]) to the standard normal variable \[Z\]. This standardization process is crucial in probability calculations, enabling us to use the standard normal distribution table to find probabilities associated with any normal distribution. To solve the problems given in the exercise, transformations of the variables to the standard form is essential.
By using the formula:- \[Z = \frac{{X - \mu}}{{\sigma}}\]We convert a normal variable \[X\] (with mean \[\mu\] and standard deviation \[\sigma\]) to the standard normal variable \[Z\]. This standardization process is crucial in probability calculations, enabling us to use the standard normal distribution table to find probabilities associated with any normal distribution. To solve the problems given in the exercise, transformations of the variables to the standard form is essential.
Mean and Variance Calculations
Calculating the mean and variance for lognormal distributions differs slightly from normal distributions due to the transformation involved. In a lognormal distribution, if \[X\] follows this distribution, then \[\ln(X)\] is normally distributed.- **Mean of a Lognormal Distribution**: Calculated with the formula: \[E(X) = e^{\mu + \frac{\sigma^2}{2}}\] This reflects the exponential growth expected in processes like stock prices.
- **Variance of a Lognormal Distribution**: Computed as: \[\text{Var}(X) = \left(e^{\sigma^2} - 1\right) e^{2\mu + \sigma^2}\] This formula demonstrates how the variance also grows exponentially, a characteristic of lognormal distributions. Understanding these calculations is vital in making informed decisions based on statistical data distributions.
- **Variance of a Lognormal Distribution**: Computed as: \[\text{Var}(X) = \left(e^{\sigma^2} - 1\right) e^{2\mu + \sigma^2}\] This formula demonstrates how the variance also grows exponentially, a characteristic of lognormal distributions. Understanding these calculations is vital in making informed decisions based on statistical data distributions.
Probability Calculations
In statistics, probability calculations involve determining how likely an event is to occur within a particular distribution. With normal and lognormal distributions, understanding these probabilities helps predict data behavior.For a lognormal variable, converting it to a normal distribution using natural logarithms allows us to employ standard normal distribution tables or integrals to find probabilities. Here鈥檚 an outline for solving given types of probability problems:- **Finding \(P(X
