Probability Density Function (PDF) of Lognormal Distribution

What is Lognormal Distribution?

The lognormal distribution is a way to describe the likelihood of different values for a variable. This variable has a special property; if logarithm (log) is taken as its value, those log values follow a normal distribution. In layman’s language, if we have a variable X that follows a lognormal distribution, then if we take the natural logarithm (ln) of X, we’ll get a normal distribution. If X has a lognormal distribution with parameters μ and σ, then, X ~ log N (μ,σ2)....

Probability Density Function (PDF) of Lognormal Distribution

The probability density function (PDF) for the lognormal distribution depends on two parameters, μ (mean) and σ (standard deviation), for x values greater than 0. When we take the logarithm of our lognormal data, μ represents the mean, and σ is the standard deviation of this transformed data....

Lognormal Distribution Curve

It is right-skewed, meaning it tilts to the right.The curve begins at zero, rises to its peak, and then declines.The degree of skewness increases as the standard deviation (σ) rises, keeping the mean (μ) constant.μ represents the mean of natural logarithms of the data.σ represents the standard deviation of natural logarithms of the data.When σ is much larger than 1, the curve rises steeply at the start, peaks early, and then falls rapidly, resembling an exponential curve.In this distribution, μ acts as more of a scale parameter, unlike the normal distribution where it serves as a location parameter....

Mean and Variance of Lognormal Distribution

Mean (μ)...

Applications of Lognormal Distribution

1. Rubik’s Cube Solving Times: The time taken by individuals to solve a Rubik’s Cube, whether by an individual or as part of a general population, often follows a lognormal distribution. This distribution can help analyse and predict solving times....

Examples of Lognormal Distribution

Example 1:...

Difference Between Normal Distribution and Lognormal Distribution

Characteristic Normal Distribution Lognormal Distribution ShapeSymmetricalRight-skewedRange of ValuesFrom negative to positiveFrom zero to positiveParameter InterpretationMean (μ) and Standard Deviation (σ)Mean of ln(x) (μ) and Standard Deviation of ln(x) (σ)Data TransformationNot transformedNatural logarithm transformation of dataApplicationsCommon in many natural phenomena such as heights, weights, IQ scoresUsed for data with positive values that exhibit right-skewed patterns, like income, stock prices, and resource reservesReal-life ExamplesHeights, weights, IQ scoresStock returns, resource reserves, income distributionProbability Density FunctionSymmetrical bell-shaped curveRight-skewed, starts from zero and rises to a peakMean and VarianceDefine the central tendency and spread of dataDefine the central tendency and spread of the natural logarithm of the dataCommon Parameter Valuesμ (mean) and σ (standard deviation)μ and σ represent parameters of the natural logarithm of the data...