Lognormal Distribution Calculator
Calculate probabilities, quantiles, and statistical measures for lognormal distributions
Distribution Parameters
Mean of ln(X); can be any real number
Standard deviation of ln(X); must be positive
Value at which to evaluate the PDF
Results
Distribution: Lognormal(μ = 0, σ = 1)
PDF Formula: f(x) = (1/(x·σ·√(2π))) · exp(-(ln(x)-μ)²/(2σ²))
Mean Formula: E[X] = exp(μ + σ²/2) = 1.6487
Stock Price Example
Financial Modeling
Scenario: Stock price follows lognormal distribution
Parameters: μ = 0.1 (scale), σ = 0.2 (shape)
Current price: $100 (normalized to x = 1)
Question: What's the probability the stock price is below $120?
Solution
For x = 1.2: CDF = Φ((ln(1.2) - 0.1) / 0.2)
CDF = Φ((0.1823 - 0.1) / 0.2) = Φ(0.4115)
P(X ≤ 1.2) ≈ 0.6596 or 65.96%
Expected value: exp(0.1 + 0.2²/2) = exp(0.12) ≈ 1.127
Key Properties
Positive Support
X > 0 (only positive values)
Log-Normal
ln(X) ~ Normal(μ, σ²)
Right Skewed
Long tail to the right
Multiplicative
Product of lognormals is lognormal
Common Applications
Stock prices and returns
Income distributions
Particle size distributions
Reliability and failure times
Internet traffic analysis
Medical dosage modeling
Understanding Lognormal Distribution
What is Lognormal Distribution?
A lognormal distribution is a continuous probability distribution where the logarithm of the random variable follows a normal distribution. If X is lognormally distributed, then ln(X) follows a normal distribution with parameters μ and σ.
Key Characteristics
- •Support: x > 0 (only positive values)
- •Right-skewed distribution
- •Heavy right tail
- •Mean > Median > Mode
Mathematical Formulas
PDF: f(x) = (1/(x·σ·√(2π))) · e^(-(ln(x)-μ)²/(2σ²))
CDF: F(x) = Φ((ln(x)-μ)/σ)
- μ: Scale parameter (mean of ln(X))
- σ: Shape parameter (std dev of ln(X))
- Φ: Standard normal CDF
- x: Value (must be positive)
Note: Parameters μ and σ refer to the underlying normal distribution of ln(X), not the lognormal distribution itself.
Distribution Measures
Central Tendency
- Mean: exp(μ + σ²/2)
- Median: exp(μ)
- Mode: exp(μ - σ²)
Dispersion
- Variance: [exp(σ²)-1]·exp(2μ+σ²)
- Std Dev: √Variance
- CV: √[exp(σ²)-1]
Shape
- Skewness: [exp(σ²)+2]·√[exp(σ²)-1]
- Always positive
- Right-skewed