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The mean and standard deviation of a lognormal distribution are unique in their calculation and differ from those of a normal distribution. Given the parameters \( \mu = 1.9 \) and \( \sigma = 0.9 \), these values describe the distribution of the natural logarithm of the data.
To find the true mean (expected value) of the distribution, we use the formula:

• \( E(X) = e^{\mu + \frac{\sigma^2}{2}} \)

Substituting the values, \( E(X) = e^{1.9 + \frac{0.9^2}{2}} = e^{2.305} \). Calculating, we find \( E(X) \approx 10.03 \). This tells us that the average concentration of \( \mathrm{SO}_2 \) is around 10.03.
The standard deviation of a lognormal distribution is calculated using:

• \( \text{SD}(X) = \sqrt{(e^{\sigma^2} - 1) \cdot e^{2\mu + \sigma^2}} \)

For the provided parameters, this becomes \( \text{SD}(X) \approx 11.17 \). The standard deviation indicates the level of variability or dispersion of the concentration values.

In working with a lognormal distribution, probabilities involve converting data into the logarithmic scale and then applying the normal distribution.
First, to find the probability that the \( \mathrm{SO}_2 \) concentration is at most 10, convert 10 into its logarithmic value. Compute:

• \( Z = \frac{\ln(10) - \mu}{\sigma} = \frac{2.302 - 1.9}{0.9} \approx 0.447 \)

Using the standard normal distribution, \( P(Z \leq 0.447) \approx 0.672 \), so there's about a 67.2% chance that the concentration is less than or equal to 10.
For concentrations ranging from 5 to 10, calculate the probability for 5, which involves:

• \( Z = \frac{\ln(5) - 1.9}{0.9} = \frac{1.609 - 1.9}{0.9} \approx -0.324 \)

The probability \( P(Z \leq -0.324) \approx 0.373 \). The probability of being between 5 and 10 is \( 0.672 - 0.373 = 0.299 \), or 29.9%.

Identifying parameters \( \mu \) and \( \sigma \) is crucial for utilizing a lognormal distribution effectively. These parameters are not measuring the raw data directly but instead the logarithm of the data.
\( \mu \) represents the mean of the log-transformed values; it indicates where the center of the log-transformed distribution is located. A value of \( \mu = 1.9 \) suggests a central position for the logarithmic transformation of \( \mathrm{SO}_2 \) concentrations.
\( \sigma \) signifies the standard deviation of the natural log transformation. At \( \sigma = 0.9 \), it reflects the spread or dispersion of these log-transformed concentrations.
Together, these parameters help convert the real-world measurements into a format where statistical methods for normal distributions apply, a necessary step before performing any probability calculations.

Understanding Lognormal Probability is essential for interpreting data that is positively skewed, where values do not go negative.
The lognormal probability distribution applies when the natural logarithm of a dataset results in a normal distribution, which is common in various natural phenomena.

• The concentration of \( \mathrm{SO}_2 \) above the forest fits this type, as suggested in the study referenced.
• Knowing that it is lognormal allows us to apply techniques and formulas from normal distributions, but only after logging the data.

Given the proper context, lognormal probability provides a meaningful way to model and predict real-world phenomena that are not symmetrical and are bound on one side.
This approach is practical when dealing with chemical concentrations, time-to-failure data, and financial returns, where typical values skew to the right.