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A lognormal distribution is a bit like a "transformed" version of a normal distribution. To find the mean and variance of our lognormal distribution, we first need to think about the related normal distribution of the variable's logarithm.
For a lognormal variable, the mean is calculated using the formula:

• \[ E[X] = e^{\theta + \frac{\omega^2}{2}} \]

If you plug in the values \( \theta = 1.5 \) and \( \omega = 0.4 \), you get:\[ E[X] = e^{1.5 + \frac{0.4^2}{2}} = e^{1.58} \]Once you compute \( e^{1.58} \), you can find the exact mean value.
Variance, on the other hand, shows how much values deviate from the mean. For the variance of a lognormal distribution, use:

• \[ \text{Var}(X) = \left( e^{\omega^2} - 1 \right) e^{2\theta + \omega^2} \]

Substitute the known values:\[ = \left( e^{0.4^2} - 1 \right) e^{3.08} \]Calculate \( e^{0.16} \) and then \( e^{3.08} \), to find the variance of our lognormal variable.

Calculating the probability \( P(X < 8) \) for a lognormal distribution involves a conversion to make things simpler. This conversion turns the lognormal problem into a normal distribution problem. If \( X \) is lognormal, then \( \ln(X) \) is normal.
First, convert to standard normal by computing:

• Find \( Z \), a standard normal variable, using the formula:

\[ Z = \frac{\ln(8) - \theta}{\omega} \]Calculate \( \ln(8) \), then plug in \( \theta = 1.5 \) and \( \omega = 0.4 \).
Find the standard normal cumulative probability for this \( Z \) value. This gives you \( P(X < 8) \), by checking a standard normal distribution table or using a calculator.

The way a lognormal distribution behaves compared to a normal distribution can be interesting. A lognormal distribution, defined only for positive values, has some key differences when it comes to probabilities related to zero.
For a lognormal distribution, \( P(X < 0) \) is always zero. This is because it's impossible for a log-transformed normal variable exponentiated back to real-life terms to drop below zero; in simpler terms, a lognormal variable can't be negative.
In contrast, a normal distribution allows for negative values. Even when a normal distribution and a lognormal distribution have the same mean and variance, \( P(X < 0) \) for the normal distribution won't be zero. The tails of a normal distribution stretch infinitely in both directions, which includes negative values.
Understanding these differences helps in choosing the correct statistical model for data analysis, ensuring realistic and accurate representation of data based on its nature.