Problem 182 - Applied Statistics and Probability for Engineers

Question

An article in Chemosphere ["Statistical Evaluations Reflecting the Skewness in the Distribution of TCDD Levels in Human Adipose Tissue" \((1987,\) Vol. \(16(8),\) pp. \(2135-\) 2140) ] concluded that the levels of 2,3,7,8 -TCDD (colorless persistent environmental contaminants with no distinguishable odor at room temperature) in human adipose tissue has a lognormal distribution (based on empirical evidence from North America). The mean and variance of this lognormal distribution in the USA are 8 and \(21,\) respectively. Let \(X\) denote this lognormal random variable. Determine the following: (a) \(P(2000

Step-by-Step Solution

Answer

(a) Use calculated \(\mu_Y\) and \(\sigma_Y^2\) to find \(P(2000

Step 1: Understand the Lognormal Distribution

For a lognormal distribution, if a random variable \( X \) is lognormally distributed, then \( Y = \log(X) \) follows a normal distribution. Given the mean \( \mu \) and variance \( \sigma^2 \) of \( X \), we must first determine the parameters \( \mu_Y \) and \( \sigma_Y^2 \) of the corresponding normal distribution \( Y \).

Step 2: Calculate Parameters for the Normal Distribution

For a lognormal distribution characterized by the mean \( m \) and variance \( v \) of \( X \), the mean \( \mu_Y \) and variance \( \sigma_Y^2 \) of \( Y \) are given by \( \mu_Y = \log\left(\frac{m^2}{\sqrt{m^2 + v}}\right) \) and \( \sigma_Y^2 = \log\left(1 + \frac{v}{m^2}\right) \). Substituting \( m = 8 \) and \( v = 21 \), we calculate: \( \mu_Y = \log\left(\frac{8^2}{\sqrt{8^2 + 21}}\right) \) and \( \sigma_Y^2 = \log\left(1 + \frac{21}{8^2}\right) \).

Step 3: Solve for Part (a) - P(2000 < X < 2500)

Knowing \( Y = \log(X) \) follows \( N(\mu_Y, \sigma_Y^2) \), to find \( P(2000 < X < 2500) \), find \( P(\log(2000) < Y < \log(2500)) \). Standardize \( Y \) to calculate the probabilities using the normal distribution table or a calculator with the calculated \( \mu_Y \) and \( \sigma_Y^2 \).

Step 4: Solve for Part (b) - Value exceeded with Probability 10%

To determine the value exceeded with probability 10%, find the 90th percentile of the lognormal distribution. This requires determining \( z_{0.9} \) from the standard normal distribution, and then calculating \( x = e^{\mu_Y + z_{0.9}\cdot\sigma_Y} \).

Step 5: Solve for Part (c) - Mean and Variance of X

The mean of a lognormal distribution is given by \( E[X] = e^{\mu_Y + \sigma_Y^2/2} \). The variance is given by \( \text{Var}[X] = (e^{\sigma_Y^2} - 1) e^{2\mu_Y + \sigma_Y^2} \). Use the calculated \( \mu_Y \) and \( \sigma_Y^2 \) to determine the mean and variance.

Key Concepts

Probability Calculation

Calculating probabilities for lognormal distributions involves a few important steps. The lognormal distribution is unique because it represents data whose logarithms are normally distributed. To calculate the probability of a random variable falling within a specific range, we transform it to the normal distribution context.
For example, to determine the probability that a random variable, denoted as \( X \), falls between 2000 and 2500 (\( P(2000 < X < 2500) \)), we first take the natural log of these values. This transforms the bounds into a range applicable for a normal distribution.

• Transform the bounds: \( Y = \log(X) \)
• Calculate \( \log(2000) \) and \( \log(2500) \)
• Use the parameters of the normal distribution and standardize \( Y \)

This allows us to use the properties of the standard normal distribution to find probabilities.

Statistical Parameters

In the context of lognormal distributions, understanding the statistical parameters is crucial. These parameters include the mean and variance of the distribution, but for lognormal distributions, they correlate to a transformed normal distribution.
The parameters \( \mu_Y \) and \( \sigma_Y^2 \) for the associated normal distribution are calculated using:

• Mean: \( \mu_Y = \log\left( \frac{m^2}{\sqrt{m^2 + v}} \right) \)
• Variance: \( \sigma_Y^2 = \log\left( 1 + \frac{v}{m^2} \right) \)

These transformations allow us to handle the lognormal data with the assumptions and tools available for normal distributions. Always ensure to use the correct mathematical expressions to transform and interpret your data accurately.

Random Variables

The concept of random variables is fundamental in probability and statistics. A random variable represents a function that assigns outcomes from a random phenomenon to numerical values. Here, \( X \) represents a lognormal random variable indicating TCDD levels in human adipose tissue.
Since \( Y = \log(X) \) follows a normal distribution when \( X \) is lognormally distributed, it helps us understand the nature and behavior of \( X \). Therefore:

• \( X \) represents the original set of data or variables measured
• \( Y = \log(X) \) transforms \( X \) into a form that simplifies analysis

This transformation ties the logarithmic relationship into analyzing the normally distributed \( Y \), allowing for standard probability computations.

Normal Distribution

The normal distribution is a cornerstone in statistics and plays a key role in analyzing lognormal distributions. This is because when a variable is lognormally distributed, its logarithm is normally distributed.
A normal distribution has a bell-shaped curve, characterized by two parameters: mean (\( \mu \)) and variance (\( \sigma^2 \)). Each point on this curve represents a possible outcome of the variable, with the curve's shape depicting the likelihood of each.

• Transform lognormal to normal: \( Y = \log(X) \)
• Mean becomes \( \mu_Y \)
• Variance becomes \( \sigma_Y^2 \)

Utilizing the normal distribution allows us statistical analysis through standard tables or software, simplifying calculations and interpretations for lognormal data.

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Problem 181

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Problem 184

Other exercises in this chapter

Problem 180

An article in Journal of Hydrology ["Use of a Lognormal Distribution Model for Estimating Soil Water Retention Curves from Particle-Size Distribution Data" \((2

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Problem 181

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Problem 184

Suppose that \(X\) has a beta distribution with parameters \(\alpha=2.5\) and \(\beta=2.5 .\) Sketch an approximate graph of the probability density function. I

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Problem 185

Suppose that \(x\) has a beta distribution with parameters \(\alpha=2.5\) and \(\beta=1 .\) Determine the following: (a) \(P(X

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Key Concepts

Probability Calculation

Calculating probabilities for lognormal distributions involves a few important steps. The lognormal distribution is unique because it represents data whose logarithms are normally distributed. To calculate the probability of a random variable falling within a specific range, we transform it to the normal distribution context.
For example, to determine the probability that a random variable, denoted as \( X \), falls between 2000 and 2500 (\( P(2000 < X < 2500) \)), we first take the natural log of these values. This transforms the bounds into a range applicable for a normal distribution.

• Transform the bounds: \( Y = \log(X) \)
• Calculate \( \log(2000) \) and \( \log(2500) \)
• Use the parameters of the normal distribution and standardize \( Y \)

This allows us to use the properties of the standard normal distribution to find probabilities.

Statistical Parameters

In the context of lognormal distributions, understanding the statistical parameters is crucial. These parameters include the mean and variance of the distribution, but for lognormal distributions, they correlate to a transformed normal distribution.
The parameters \( \mu_Y \) and \( \sigma_Y^2 \) for the associated normal distribution are calculated using:

• Mean: \( \mu_Y = \log\left( \frac{m^2}{\sqrt{m^2 + v}} \right) \)
• Variance: \( \sigma_Y^2 = \log\left( 1 + \frac{v}{m^2} \right) \)

These transformations allow us to handle the lognormal data with the assumptions and tools available for normal distributions. Always ensure to use the correct mathematical expressions to transform and interpret your data accurately.

Random Variables

The concept of random variables is fundamental in probability and statistics. A random variable represents a function that assigns outcomes from a random phenomenon to numerical values. Here, \( X \) represents a lognormal random variable indicating TCDD levels in human adipose tissue.
Since \( Y = \log(X) \) follows a normal distribution when \( X \) is lognormally distributed, it helps us understand the nature and behavior of \( X \). Therefore:

• \( X \) represents the original set of data or variables measured
• \( Y = \log(X) \) transforms \( X \) into a form that simplifies analysis

This transformation ties the logarithmic relationship into analyzing the normally distributed \( Y \), allowing for standard probability computations.

Normal Distribution

The normal distribution is a cornerstone in statistics and plays a key role in analyzing lognormal distributions. This is because when a variable is lognormally distributed, its logarithm is normally distributed.
A normal distribution has a bell-shaped curve, characterized by two parameters: mean (\( \mu \)) and variance (\( \sigma^2 \)). Each point on this curve represents a possible outcome of the variable, with the curve's shape depicting the likelihood of each.

• Transform lognormal to normal: \( Y = \log(X) \)
• Mean becomes \( \mu_Y \)
• Variance becomes \( \sigma_Y^2 \)

Utilizing the normal distribution allows us statistical analysis through standard tables or software, simplifying calculations and interpretations for lognormal data.