Why is the Lognormal Distribution Skewed?

The lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. This concept can be deeply understood through the interplay between normal and lognormal distributions, and the mathematical properties associated with these distributions. In this article, we explore the fundamental reasons behind the skewness of lognormal distributions, using Jensen's Inequality and logarithmic transformations.

Introduction to Normal Distribution and Logarithmic Transformation

Let us first establish the foundation by understanding the normal distribution. A random variable (X sim N(mu, sigma^2)) represents a normally distributed random variable with mean (μ) and variance (σ^2). Now, consider the transformation of this random variable through the logarithmic function. Specifically, let (Y e^X). The function (x mapsto e^x) is order-preserving, meaning that if (x_1 , then (e^{x_1} . Therefore, the median of (Y) is (e^μ) since the median of a normally distributed random variable is the mean (μ).

Application of Jensen's Inequality

Next, we utilize Jensen's Inequality, which helps us understand the expected value of a function applied to a random variable. For a convex function (f) and a random variable (X), Jensen's Inequality states that: [mathbb{E}[f(X)] geq f(mathbb{E}[X])]

Applying this to our case, let (f(X) e^X), which is a convex function. Then: [ mathbb{E}[Y] mathbb{E}[e^X] geq e^{mathbb{E}[X]} e^μ ]

The inequality indicates that the expected value of (Y) is greater than or equal to (e^μ). This can be interpreted as the right tail of the distribution of (Y) being heavier, or in other words, the lognormal distribution is skewed to the right. This explains why the transformation to a logarithmic scale introduces skewness to the distribution.

Understanding the Logarithmic Function and Real Solutions

The logarithmic function, specifically when using base (e), is the natural logarithm. It is defined as (log_e X), which is the value of (Y) that satisfies the equation (e^Y X). This function has several important properties:

For positive real numbers (X), the equation has a real solution. For zero and negative non-integer values of (X), the equation has imaginary solutions. Negative integer values and 0 give indeterminate infinite solutions.

For the lognormal distribution, (B e), and (Y) is normally distributed. Therefore, (Y) is a real number, and (X e^Y) is the observed variate. The condition that (Y) is real implies that (X) must be positive. This is why the lognormal distribution is skewed to the right, as it cannot account for zero or negative values.

Conclusion and Further Exploration

In conclusion, the skewness of the lognormal distribution arises due to the properties of the exponential function and the application of Jensen's Inequality. These mathematical properties highlight the fundamental nature of the lognormal distribution and its applications in various fields, including finance, biology, and engineering. To further explore these concepts, one might consider studying the behavior of the lognormal distribution under different transformations and understanding how skewness can be utilized in practical scenarios.