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Probability calculations are a crucial aspect when working with lognormal distributions. We use probabilities to determine the likelihood of certain events happening, such as whether a concentration will be below a certain threshold. For a lognormal distribution, it is important to understand that the natural logarithm of the variable is normally distributed. This is why calculations often involve converting values into their logarithmic equivalents.

In this exercise, we calculate the probability of \(X\), the concentration of \(\mathrm{SO}_2\), being at most 10. This probability is given by converting the value 10 into its natural logarithm, \(\ln(10)\), which approximately equals 2.302. Subsequently, this log value is plugged into the standard normal distribution formula:

• Calculate the Z-score: \Z = \frac{\ln(10) - \mu}{\sigma}\
• Find the corresponding probability using standard normal distribution tables
• Substitute known values: \Z = \frac{2.302 - 1.9}{0.9} \approx 0.4477\
• Probability: \P(Z \leq 0.4477) \approx 0.6723\

Similarly, to find the probability between two values (5 and 10), you determine the probability of the concentration being below each and subtract the smaller from the larger probability. This helps in understanding the range where the concentration usually lies.

Mean value estimation in a lognormal distribution requires a different approach compared to a normal distribution. The mean (or expected value) of a lognormal variable is not the simple average but a function of its parameters \(\mu\) and \(\sigma\). In mathematical terms, the mean is found using the formula:

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

This formula incorporates both the mean () and the variance (^2) of the underlying normal distribution. For our example with \(\mu = 1.9\) and \(\sigma = 0.9\), the mean is calculated as follows:

• Substitute the values into the formula: \E(X) = e^{1.9 + \frac{0.9^2}{2}} = e^{2.305}\
• Evaluate the exponential: This calculates to approximately 10.03

It is crucial to comprehend that this mean represents the typical concentration level of \(\mathrm{SO}_2\) under the given conditions, reflecting the average expected value when considering the entire distribution.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. For lognormal distributions, it indicates how much \(\mathrm{SO}_2\) concentration levels are likely to deviate from the mean. In mathematical terms, the standard deviation for a lognormal distribution is calculated using:

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

This formula considers both the variability of the logarithmic scale and the exponential transformation.

To find the standard deviation for our distribution with \(\mu = 1.9\) and \(\sigma = 0.9\), we substitute these values:

• Evaluate exponential parts: \(e^{\sigma^2} \approx e^{0.81}\)
• Combine to find standard deviation: \SD(X) = \sqrt{(e^{0.81} - 1) e^{4.61}}\
• Result approximately equals 10.04

Understanding standard deviation helps in grasping how much the concentration can spread around the mean. Larger standard deviations indicate a wider spread of concentration values, whereas smaller values hint at concentrations typically clustering near the mean.