gen-state-the-normal-distribution-precisely-and-explain-why-it-m-fe89

State the normal distribution precisely and explain why it matters in 2-3 sentences.

# Normal Distribution **Precise Definition:** A random variable X follows a normal distribution N(μ, σ²) if its probability density function is: $$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$ where μ is the mean and σ² is the variance. **Why It Matters:** The normal distribution is fundamental because the Central Limit Theorem guarantees that sums and averages of independent random variables converge to it, making it arise naturally in countless real-world phenomena. This property underpins most of statistical inference, from hypothesis testing to confidence intervals, allowing us to make probabilistic statements about data even when individual observations aren't normally distributed.