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A normal distribution is one of the most common and well-known statistical distributions. It is described by a bell-shaped curve, often referred to as the Gaussian distribution. The main characteristics of a normal distribution include:

• Symmetrical shape around its mean.
• Total area under the curve equals 1, representing total probability.
• Densely clustered values around the mean, with fewer occurrences further away.

In this exercise, we begin by considering the transformation of a lognormal variable, thereby focusing on the properties of its natural logarithm, which follows a normal distribution. The given lognormal variable X has a corresponding log-transformed variable Y. Here, the parameter values given, \(\theta = 5\) and \(\omega^2 = 9\), define the mean and variance of the normal distribution for Y as: Y follows \(N(5, 3)\) where 3 is the standard deviation calculated as the square root of the variance.

Calculating probabilities under a distribution curve is key to understanding statistical patterns. With a normal distribution, we often calculate probabilities by "standardizing" a variable. This involves transforming it into a standard normal distribution with mean 0 and variance 1 using a z-score.
To determine \(P(X < 13,300)\) for our lognormal distribution, we first compute \(P(\ln X < \ln 13,300)\). The value \(\ln 13,300\) serves as the threshold when transformed to the normal scale. This produces a z-score: \[Z = \frac{\ln 13,300 - 5}{3}\].After standardizing, the probability is easily obtained using a standard normal distribution table or technology tools, which yields \(P(Z < 1.4991) \approx 0.9332\). This result highlights that there's about a 93.32% chance that the lognormal variable X is less than 13,300.

Understanding the properties of distributions aids in interpreting the behavior of random variables. A lognormal distribution like the one in this problem is asymmetrical, differing from the normal distribution's symmetry. Its shape is commonly right-skewed due to the exponential nature of its development from a normally distributed variable.

• The parameter \(\theta\) affects the location, akin to the mean in the transformed normal distribution.
• The parameter \(\omega^2\) determines the spread or variation.
• Unlike a linear transformation, the back-transformation to linear scale increases skewness.

Despite these differences, by understanding the underlying or transformed normal distribution, we gauge the spread and probability statements using normal distribution simplifications.

Determining the mean and variance of a lognormal distribution requires specific transformations due to its exponential nature.
For this exercise, the mean of X, denoted as \(\mu_X\), is calculated using the formula \(\mu_X = e^{\theta + \frac{\omega^2}{2}}\). Plugging in \(\theta = 5\) and \(\omega^2 = 9\), we obtain \(\mu_X = e^{9.5}\), providing an approximate mean of 13,399.93.
The variance, on the other hand, is derived with the formula \(\sigma_X^2 = (e^{\omega^2} - 1)e^{2\theta + \omega^2}\). This combines both parameters to determine spread on a logarithmic scale. Calculation methods reveal a substantial variance: roughly 1.446 x 10^{12}. These values highlight how unique parameters substantially adjust the mean and variance of lognormal compared to normal distributions.