Introduction:

Probability distributions serve as essential tools in statistical modeling, helping us understand the likelihood of different outcomes. Let's unravel the intricacies of Uniform, Log-Normal, and Pareto Distributions:

1. Uniform Distribution:

a. Definition:

• • A Uniform Distribution represents outcomes where each value in the range is equally likely.

b. PDF and CDF Equations:

• • PDF: ( \(f(x) = \frac{1}{b-a} \) ), where (a) and (b) are the lower and upper bounds.

    • CDF: ( \(F(x) = \frac{x-a}{b-a}\) ) for (a \leq x \leq b), 0 otherwise.

c. Parameters:

• • (a) and (b) denote the lower and upper bounds.

d. Example Usage:

• • Modeling outcomes where each value in a given range is equally probable, like rolling a fair die.

2. Log-Normal Distribution:

a. Definition:

• • The Log-Normal Distribution arises from the exponential transformation of a normal distribution.

b. PDF and CDF Equations:

• • PDF: ( \(f(x) = \frac{1}{x\sigma \sqrt{2\pi}} e^{-\frac{(\ln(x) - \mu)^2}{2\sigma^2}}\) )

    • CDF: No closed-form expression; often computed numerically.

c. Parameters:

• • ( \(\mu\)) is the mean of the natural logarithm of the distribution, ( \(\sigma\) ) is the standard deviation.

d. Example Usage:

• • Modeling stock prices, where prices are influenced by multiplicative factors.

3. Pareto Distribution:

a. Definition:

• • The Pareto Distribution represents power-law distributions where a small number of items account for the majority of the distribution.

b. PDF and CDF Equations:

• • PDF: ( \(f(x) = \frac{\alpha x_m^\alpha}{x^{\alpha+1}}\) )

    • CDF: ( \(F(x) = 1 - \left(\frac{x_m}{x}\right)^\alpha\) )

c. Parameters:

• • ( \(\alpha\)) is the shape parameter, ( \(x_m\) ) is the minimum value.

d. Example Usage:

• • Modeling wealth distribution, where a small percentage of the population holds the majority of resources.

Conclusion:

Understanding these distributions broadens our toolkit for modeling diverse real-world scenarios. From the evenness of a Uniform Distribution to the skewed wealth distribution captured by Pareto, each distribution serves a unique purpose in statistical analysis.