A lognormal random variable is one whose logarithm is normally distributed. Lognormal distributions arise naturally when the effect of many small independent forces is multiplicative. This contrasts with the normal distribution, which arises naturally when many small independent forces act additively.
The function Φ(x) in the cdf denotes the standard normal cdf function, which has the property that Φ(-x) = 1 - Φ(x).
| Parameter | Range | Description |
|---|---|---|
| μ | −∞ < μ < ∞ | Mean of associated normal distribution |
| σ² | σ² > 0 | Variance of associated normal distribution |
Probability Density Function
Support
Mean
Variance
| Example | μ | σ² |
|---|---|---|
| The logarithm of the price of individual stocks relative to one year earlier is normally distributed with mean 0.07 and variance 0.85. | 0.0700 | 0.8500 |
| Response times for tasks are often modeled by a lognormal distribution. Let X be the response time of a random person on a task that takes an average 60 seconds with standard deviation 30 seconds. | 3.983 | 0.2231 |
| The body weight of adult women in the US is approximately lognormally distributed with mean 170.6 pounds and standard deviation 24.98 pounds. | 5.129 | 0.0212 |
X ∼ Lognormal(μ, σ²)
E(X) = , Var(X) =
Note that, unlike the normal distribution, the pdf of the lognormal is skewed to the right, with a long right tail. This often makes it a good model for variables which are bounded below but not above.
Note that, unlike the normal distribution, the pdf of the lognormal is skewed to the right, with a long right tail. This often makes it a good model for variables which are bounded below but not above.
The graph above displays the survival function S(x) = P(X > x) = 1 - F(X), where F(x) is the cumulative distribution function (cdf).
Survival functions are used in survival analysis, a branch of statistics concerned with the expected duration until an event occurs such as death or the failure of a mechanical system.
The graph above displays the hazard function h(x). This equals f(x)/S(x), where f(x) is the pdf and S(x) = P(X > x) is the survival function.
Note that, unlike the hazard function for the normal(μ, σ2) distribution which steadily increases, the lognormal hazard function is unimodal with a long right tail.
The illustration above shows a random variable N with a normal(μ, σ2) distribution in the lower left corner. The random variable X = eN in the upper right has a lognormal(μ, σ2) distribution. The quartiles are shown for both distributions. Note that the function eN stretches the gap between quartiles as N increases, transforming the symmetric normal distribution into the right-skewed lognormal.
The simulation above uses a point N chosen at random from a normal(μ, σ2) distribution. The light blue line shows the value of the random variable X = eN, which has a lognormal(μ, σ2) distribution. The histogram accumulates the results of each simulation.
Y =
E(Y) = , Var(Y) =