Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is
f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0
where ln x has zero mean and standard deviation σ > 0.
b. Evaluate lim x → 0 ƒ(x). (Hint: Let x = eʸ.)

Verified step by step guidance

1. Start by writing down the given probability density function (pdf) for the log-normal distribution: \(f(x) = \frac{1}{x \sigma \sqrt{2\pi}} e^{-\frac{(\ln x)^2}{2 \sigma^2}}\), for \(x \geq 0\).
2. To evaluate the limit as \(x\) approaches 0, use the hint and substitute \(x = e^y\). This means as \(x \to 0^+\), we have \(y = \ln x \to -\infty\).
3. Rewrite the function in terms of \(y\): \(f(e^y) = \frac{1}{e^y \sigma \sqrt{2\pi}} e^{-\frac{y^2}{2 \sigma^2}} = \frac{1}{\sigma \sqrt{2\pi}} e^{-y} e^{-\frac{y^2}{2 \sigma^2}}\).
4. Combine the exponents to get a single exponential expression: \(f(e^y) = \frac{1}{\sigma \sqrt{2\pi}} e^{-y - \frac{y^2}{2 \sigma^2}}\).
5. Analyze the behavior of the exponent \(-y - \frac{y^2}{2 \sigma^2}\) as \(y \to -\infty\). This will help determine the limit of \(f(x)\) as \(x \to 0^+\).

Key Concepts

Log-normal Distribution

A log-normal distribution describes a random variable whose logarithm is normally distributed. If Y = ln(X) follows a normal distribution, then X is log-normally distributed. This distribution is skewed and only defined for positive values, commonly used in modeling multiplicative processes.

Limit Evaluation Using Substitution

Evaluating limits involving complex functions often requires substitution to simplify the expression. Here, substituting x = e^y transforms the limit as x approaches 0 into a limit as y approaches -∞, making it easier to analyze the behavior of the function near zero.

Behavior of Exponential and Logarithmic Functions at Infinity

Understanding how exponential and logarithmic functions behave as their arguments approach infinity or negative infinity is crucial. For example, e^y approaches 0 as y → -∞, and ln(x) approaches -∞ as x → 0+, which helps in evaluating limits involving these functions.