The length of time (in seconds) that a user views a page on...

The length of time (in seconds) that a user views a page on a Web site before moving to another page is a lognormal random variable with parameters $\theta=0.5$ and $\omega^{2}=1$. (a) What is the probability that a page is viewed for more than 10 seconds? (b) By what length of time have $50 \%$ of the users moved to another page? (c) What are the mean and standard deviation of the time until a user moves from the page?

The length of time (in seconds) that a user views a page on a Web site before moving to another page is a lognormal random variable with parameters $\theta=0.5$ and $\omega^{2}=1$.
(a) What is the probability that a page is viewed for more than 10 seconds?
(b) By what length of time have $50 \%$ of the users moved to another page?
(c) What are the mean and standard deviation of the time until a user moves from the page?

Key Concepts

Lognormal Distribution

The lognormal distribution is a probability distribution of a random variable whose logarithm is normally distributed. It is often used to model variables that are positive and skewed, such as time durations, incomes, or biological measurements. When a variable is said to be lognormal with parameters, these parameters are associated with the underlying normal distribution of the logarithm of the variable.

Transformation to Normal Distribution

Since the natural logarithm of a lognormally-distributed variable follows a normal distribution, problems involving probabilities or quantiles of a lognormal variable can be transformed into problems involving the standard normal distribution. This transformation simplifies calculations by allowing the use of standard normal tables or software functions for the cumulative distribution function (CDF) of the bell curve.

Percentile Calculation in Lognormal Distribution

The median or any quantile of a lognormal distribution is calculated by taking the equivalent quantile in the normal distribution for the logarithm of the variable, and then exponentiating the result. This process leverages the relationship between the lognormal and normal distributions, enabling the determination of specific cutoff points, like the 50th percentile (median), for the original variable.

Moments of Lognormal Distribution

The moments of a lognormal distribution, such as the mean and variance (or standard deviation), are computed using formulas that involve the parameters of the underlying normal distribution. Specifically, the mean is given by the exponential of the sum of the location parameter and half the variance, while the variance incorporates an additional exponential term that reflects the spread of the underlying normal distribution. These formulas highlight the impact of the normal distribution parameters on the scale and spread of the lognormal variable.

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