The length of time, in seconds, that a computer user takes...

The length of time, in seconds, that a computer user takes to read his or her e-mail is distributed as a lognormal random variable with $\mu=1.8$ and $\sigma^{2}=4.0$. (a) What is the probability that a user reads e-mail for more than 20 seconds? More than a minute? (b) What is the probability that a user reads e-mail for a length of time that is equal to the mean of the underlying lognormal distribution?

The length of time, in seconds, that a computer user takes to read his or her e-mail is distributed as a lognormal random variable with $\mu=1.8$ and $\sigma^{2}=4.0$.
(a) What is the probability that a user reads e-mail for more than 20 seconds? More than a minute?
(b) What is the probability that a user reads e-mail for a length of time that is equal to the mean of the underlying lognormal distribution?

Key Concepts

Standardization and Z-Scores

Standardization is the process of converting a normal random variable to a standard normal variable (z), which has a mean of 0 and a variance of 1. This is done by subtracting the mean and dividing by the standard deviation, enabling the use of cumulative distribution function (CDF) tables or software to find probabilities for specific events in the normal (and thus the transformed lognormal) setting.

Mean of a Lognormal Distribution

The mean of a lognormal distribution is given by exp(mu + sigma^2/2), which generally exceeds the median (exp(mu)) because of the distribution's skewness. This concept is crucial when interpreting results from a lognormal model, especially in contexts where the average behavior of the variable is compared to individual observations.

Cumulative Distribution Function (CDF) and Probability Calculations

The cumulative distribution function (CDF) is a function used to determine the probability that a random variable takes on a value less than or equal to a given threshold. For both the normal and lognormal distributions, the CDF is instrumental in calculating the probabilities of events that span certain intervals of the variable.

Transformation to the Normal Distribution

A key property of the lognormal distribution is that taking the natural logarithm of a lognormal random variable yields a variable that follows a normal distribution. This transformation is pivotal because it allows one to apply the well-developed theory and tools of the normal distribution, such as z-scores and standard normal tables, to compute probabilities.

Lognormal Distribution

The lognormal distribution describes a random variable whose logarithm has a normal distribution. It is useful for modeling data that are positively skewed and are naturally constrained to be nonnegative, such as time durations, sizes, or incomes. Its parameters, often denoted as mu and sigma, describe the mean and variance of the logarithm of the underlying variable.

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