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The lognormal distribution provides a unique approach in determining the mean and standard deviation because it is inherently transformed from a normal distribution. This makes the calculations quite interesting! Instead of handling the data outright, the data is transformed to a normal distribution via a natural logarithm. This means if you have a variable \(X\) that follows a lognormal distribution, then \(Y = \ln(X)\) would follow a normal distribution with the given parameters \(\mu\) and \(\sigma\).

For our \(\text{SO}_2\) concentration, the mean \(E(X)\) is computed using the formula for the mean of a lognormal distribution: \[E(X) = e^{\mu + \frac{\sigma^2}{2}}\] Plug in \(\mu = 1.9\) and \(\sigma = 0.9\) to get \(E(X) \approx 8.395\).
This value indicates the typical or average concentration of \(\text{SO}_2\) above the forest when the data is transformed back from the logarithmic scale.

Calculating the standard deviation put it one step further, using:\[\sigma_X = e^{\mu + \frac{\sigma^2}{2}} \sqrt{e^{\sigma^2} - 1}\]With the same parameter values, you find this to be \(\sigma_X \approx 8.671\).
This showcases the dispersion or spread of \(\text{SO}_2\) concentrations around that mean. The larger the standard deviation, the wider the range of values can be expected.

The cumulative distribution function (CDF) of a lognormal distribution plays a crucial role when you need to find probabilities of where a variable might fall. The CDF helps assess the probability that a variable is less than or equal to a given value. For the lognormal distribution, this involves transforming the value so it fits within the standard normal distribution framework.

For a \(\text{SO}_2\) concentration of 10, you'd calculate \(P(X \leq 10)\) by first finding \(Y = \ln(10)\).
Then, standardize this by computing \(\frac{\ln(10) - \mu}{\sigma}\), where \(\mu = 1.9\) and \(\sigma = 0.9\).
This gives you approximately 0.534, and the next step involves using a standard normal distribution table to find \(P(Z \leq 0.534)\), resulting in a value around 0.703.

This means there's a 70.3% probability that the \(\text{SO}_2\) concentration is at or below 10. In probability terms, a lower CDF value indicates lower concentration limits, while a higher value covers broader ranges.

Probability calculations help us understand the chances of an event happening within certain bounds. For our \(\text{SO}_2\) concentration, you're looking at two aspects: the chance of concentrations being at most 10 and between 5 and 10.

Calculating probabilities between two values comes by subtracting one CDF value from another.
From previous calculations, \(P(X \leq 10)\) was found to be approximately 0.703. Next, we calculate \(P(X \leq 5)\) by standardizing it to \(\frac{\ln(5) - 1.9}{0.9}\), which is approximately -0.563.
This process provides a CDF of 0.287 for a concentration of 5. To find the probability of concentrations between 5 and 10, you calculate \(0.703 - 0.287\), which is 0.416.

This result reflects a 41.6% probability that the \(\text{SO}_2\) concentration falls within this range. By understanding how to find these probabilities, you can easily determine ranges and likelihoods for any given set of data under a lognormal distribution.