Question

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.
e. For what value of σ > 0 in part (d) does ƒ(x*) have a minimum?

Accepted Answer

First, write down the given probability density function (pdf) of 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. $$Identify the variable with respect to which you want to find the minimum of $$f(x^*). $$Here, $$x^* is $$fixed, and you want to find the value of $$\sigma \u003e 0$$ that minimizes $$f(x^*). $$Treat $$f(x^*) as a $$function of $$\sigma$$ only:  
$$f(\sigma) = \frac{1}{x^* \sigma \sqrt{2\pi}} e$$^{-\frac{(\ln x^*)^2}$${2 \$$sigma^2$$}}. To $$find the minimum, compute the derivative of $$f(\sigma)$$ with respect to $$\sigma$$, denoted $$f'(\sigma)$$, using the product and chain rules. Remember to differentiate both the $$1/\sigma$$ term and the exponential term. Set the derivative $$f'(\sigma) = 0$$ and solve for $$\sigma \u003e 0. $$This will give the critical points. Then, verify which critical point corresponds to a minimum by checking the second derivative or analyzing the behavior of $$f(\sigma).$$

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Key Concepts

Log-normal Distribution

Probability Density Function (PDF)

Optimization of Functions with Respect to Parameters