Problem 178 - Applied Statistics and Probability for Engineers

Question

Suppose that \(X\) has a lognormal distribution with parameters \(\theta=10\) and \(\omega^{2}=16\). Determine the following: (a) \(P(X<2000)\) (b) \(P(X>1500)\) (c) Value exceeded with probability 0.7

Step-by-Step Solution

Answer

(a) \(P(X < 2000) \approx 0.275\); (b) \(P(X > 1500) \approx 0.749\); (c) Value exceeded with probability 0.7 is approximately 2693.

Step 1: Understanding Lognormal Distribution

The lognormal distribution of a variable \(X\) with mean \(\theta\) and variance \(\omega^2\) implies that \(\ln(X)\) is normally distributed with mean \(\mu = \theta\) and variance \(\sigma^2 = \omega^2\). Therefore, \(\ln(X)\) is \(N(10, 16)\).

Step 2: Translating to Standard Normal

To solve these problems, convert \(\ln(X)\) to a standard normal variable \(Z\) using the formula: \[Z = \frac{\ln(X) - \mu}{\sigma} = \frac{\ln(X) - 10}{4}\] since the standard deviation \(\sigma\) is the square root of the variance, \(4\).

Step 3: Calculating P(X < 2000)

For part (a), we want \(P(X < 2000)\). Compute \(\ln(2000)\) first, then standardize: \[Z = \frac{\ln(2000) - 10}{4}\] Approximating \(\ln(2000) \approx 7.6\), so \(Z = \frac{7.6 - 10}{4} = -0.6\). Use a standard normal distribution table to find \(P(Z < -0.6)\).

Step 4: Calculating P(X > 1500)

For part (b), we compute \(P(X > 1500)\). So, find \(Z = \frac{\ln(1500) - 10}{4}\). Approximate \(\ln(1500) \approx 7.3\), thus \(Z = \frac{7.3 - 10}{4} = -0.675\). Use a standard normal distribution table to find \(P(Z > -0.675)\), which is equal to \(1 - P(Z < -0.675)\).

Step 5: Finding Value Exceeded With Probability 0.7

For part (c), find the value \(x\) such that \(P(X > x) = 0.7\), which is equivalent to \(P(X < x) = 0.3\). Find \(Z = -0.524\) where it corresponds to \(P(Z < -0.524) = 0.3\) in the standard normal table. Solve for \(x\): \[\ln(x) = 4Z + 10\] \[\ln(x) = 4(-0.524) + 10 = 7.904\] Convert back to \(x\): \[x = e^{7.904} \approx 2693\].

Key Concepts

Standard Normal Distribution

The standard normal distribution is a fundamental concept in statistics. It forms the backbone of many probability calculations.
The standard normal distribution, denoted as \( N(0,1) \), has a mean of 0 and a standard deviation of 1. It is symmetrically bell-shaped with 68% of the area under the curve falling within one standard deviation from the mean, 95% within two, and 99.7% within three.
When any normal distribution is converted into a standard normal distribution, it simplifies complex calculations by mapping scores on a common scale.To transform a given normal variable \( X \) with mean \( \mu \) and standard deviation \( \sigma \) into a standard normal variable \( Z \), the formula is as follows:

• \[ Z = \frac{(X - \mu)}{\sigma} \]

This transformation is crucial for interpreting probability calculations and involves translating any normally distributed data into a standard form, making statistical inference more uniform and accessible.

Probability Calculations

Probability calculations are a key part of understanding statistical distributions. When dealing with normal distributions, probabilities represent the likelihood of a random variable falling below, above, or between certain values.
The standard normal distribution has designated tables, often called Z-tables, which provide the cumulative probability of a standard normal variable \( Z \) being less than a specified value. This is invaluable for deriving probabilities related to real-world data.For instance, to compute \( P(X < 2000) \) for a lognormal distribution, we first convert \( X \) to a standard normal variable \( Z \), using logarithms:

• Calculate the logarithm of the desired value: \( \ln(2000) \)
• Standardize using \[ Z = \frac{\ln(2000) - 10}{4} \]
• Locate \( P(Z < -0.6) \) using the Z-table

This procedure allows us to transform the problem into one already mapped in standard distribution, thus facilitating precise calculation of probabilities.

Statistical Distribution Transformations

Statistical distribution transformations are essential methods for analyzing data that follows distributions other than the standard normal distribution, like the lognormal distribution. A lognormal distribution implies that while the data itself is skewed, its logarithm follows a normal distribution.
To solve problems related to lognormal distributions, we often perform transformations to convert skewed data into a normal form. This linearizes data, making it amenable to familiar statistical techniques.
The transformation process involves:

• Finding the natural logarithm of the variable, converting it into a normal distribution \( \ln(X) \)
• Standardizing this normal variable using its respective mean and standard deviation
• Using these standardized forms to compute probabilities and percentiles

For example, to find a value \( x \) such that \( P(X < x) = 0.3 \), we find the corresponding \( Z \) value from the normal distribution table, inverse transform it to \( \ln(x) \), and then exponentiate to obtain \( x \). These transformations bridge the gap between complex real-world distributions and the simpler form of normal distribution.