📘 Lognormal Distribution

🔍 What Is It?

A lognormal distribution describes a random variable whose natural logarithm is normally distributed. If

\[ W \sim \mathcal{N}(\theta, \omega^2), \quad \text{then } X = e^W \sim \text{Lognormal}(\theta, \omega) \]

• Used when values grow or decay over time and are always positive
• Common in: income, component life, wait times, insurance claims

📐 Key Formulas

• PDF: \[ f(x) = \frac{1}{x \omega \sqrt{2\pi}} \exp\left( -\frac{(\ln x - \theta)^2}{2\omega^2} \right), \quad x > 0 \]
• Mean: \( \mathbb{E}(X) = e^{\theta + \omega^2/2} \)
• Variance: \[ \text{Var}(X) = e^{2\theta + \omega^2} \left(e^{\omega^2} - 1\right) \]

📊 Lognormal vs Normal

🧮 Step-by-Step Problem

Example: A student’s battery life follows \( \text{Lognormal}(3.5, 0.4) \). What is the probability it lasts more than 40 hours?

1. We want \( P(X > 40) \)
2. Use: \[ P(X > x) = 1 - \Phi\left( \frac{\ln(x) - \theta}{\omega} \right) \]
3. Compute Z: \[ Z = \frac{\ln(40) - 3.5}{0.4} = \frac{3.688 - 3.5}{0.4} = 0.47 \]
4. Lookup \( \Phi(0.47) ≈ 0.6808 \), so: \[ P(X > 40) = 1 - 0.6808 = 0.3192 \]

✅ Answer: 31.92%

⚙️ Try It Yourself

🧪 Practice Problem

Scenario: Battery life is lognormal with \( \theta = 3.2, \omega = 0.5 \). What is \( P(X > 30) \)?