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When dealing with lognormal distributions, probability calculations often revolve around understanding the transformation that links a lognormal random variable to a normal distribution. To find probabilities for specific intervals of a lognormally distributed variable like determining \(P(500 < X < 1000)\), you start by transforming these bounds using natural logarithms. This is because a variable \(X\) is lognormally distributed if \(\ln(X)\) is normally distributed. The given parameters \(\theta = -2\) and \(\omega^2 = 9\) indicate that \(\ln(X)\) follows a normal distribution with a mean of \(-2\) and a standard deviation of \(3\) (since the standard deviation is the square root of the variance).

• Calculate \(\ln(500)\) and \(\ln(1000)\).
• Find the corresponding Z-scores using the formula \(Z = \frac{\ln(x) - \theta}{\omega}\).
• Use standard normal distribution tables to compute \(P(Z_1 < Z < Z_2)\), which gives the probability for the desired interval.

This method highlights that understanding the relationship between the lognormal distribution and its normal counterpart is essential in solving such probability problems.

The mean of a lognormal distribution, denoted as \(E(X)\), is slightly more complex than regular distributions due to the exponential transformation involved. For a lognormal distribution with parameters \(\theta\) and \(\omega^2\), the mean is calculated using the formula: \[ E(X) = e^{\theta + \frac{\omega^2}{2}} \] This formula arises because the natural logarithm of a lognormally distributed random variable results in a normally distributed variable. Therefore, the exponential factor accounts for both the mean and half of the variance of the associated normal distribution. Inserting the values from the exercise:

• With \(\theta = -2\) and \(\omega^2 = 9\), the mean is \(e^{-2 + \frac{9}{2}} = e^{2.5} \).
• This computation reflects how the variance of \(\ln(X)\) influences the location and dispersion of \(X\).

Understanding this concept is crucial for anyone looking to apply lognormal distributions in fields like finance or real-world modeling.

The variance of a lognormal distribution characterizes the spread of the distribution and can be derived from its underlying normal distribution parameters. The formula for the variance \(Var(X)\) of a lognormal distribution is: \[ Var(X) = \left(e^{\omega^2} - 1\right) e^{2\theta + \omega^2} \] This equation considers the original parameters and provides an exponentially weighted spread measure. Here's how it's applied in the context of the problem:

• Insert \(\theta = -2\) and \(\omega^2 = 9\) into the formula.
• First, compute \(e^{9}\) and subtract one from it.
• Then multiply this result by \(e^{2 \cdot (-2) + 9}\).

This systematically reveals the variance, which is usually larger due to the exponential nature of the lognormal distribution. This significant variance indicates that the distribution is skewed with a long tail on the right, which is a hallmark of lognormal behavior.