# [[Normal Distribution]] (Mathematics > Statistics) ## Definition The **normal distribution** is a continuous probability distribution that is symmetric about its mean, showing that data near the mean are more frequent than data far from the mean. It is defined by the probability density function: $ f(x)=\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}, $ where $\mu$ is the mean and $\sigma$ is the standard deviation. We denote this as: $ X \sim N(\mu, \sigma^2). $ ## Key Concepts - **Symmetry:** The distribution is symmetric about the mean $\mu$. - **Bell Curve:** The graph of the pdf forms a characteristic bell shape. - **Parameters:** Defined by the mean ($\mu$) and variance ($\sigma^2$). - **Continuous Variable:** It applies to variables taking any real value in $(-\infty, \infty)$. ## Important Properties 1. **Total Area:** The total area under the curve is 1. 2. **Inflection Points:** The curve has inflection points at $x = \mu \pm \sigma$. 3. **Standard Normal Case:** When $\mu=0$ and $\sigma=1$, the distribution is called the standard normal distribution. ## Essential Formulas - **Probability Density Function:** $ f(x)=\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}. $ - **Standardization Formula:** $ Z = \frac{X-\mu}{\sigma}. $ ## Core Examples 1. **Basic Example:** For a standard normal variable $Z \sim N(0,1)$, the pdf simplifies to: $ f(z)=\frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}z^2}. $ 2. **Advanced Example:** Given a normal variable $X \sim N(10, 4)$, standardize it using: $ Z = \frac{X-10}{2}, $ and then use standard normal tables or software to compute probabilities for $X$. ## Related Theorems/Rules - **Central Limit Theorem:** States that the sum of a large number of independent random variables tends toward a normal distribution, regardless of the original distribution. - **Empirical Rule:** Approximately 68% of the data lies within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations of the mean. ## Common Pitfalls - **Parameter Confusion:** Mistaking the standard deviation ($\sigma$) for the variance ($\sigma^2$). - **Overuse of Normality:** Applying normal distribution assumptions to data that are not normally distributed. - **Standardization Errors:** Incorrectly computing the standard normal variable $Z$. ## Related Topics - [[Standard Normal Distribution]] - [[Discrete Random Variables]] - [[Expected Value of Discrete Random Variables]] - [[Cumulative Distribution Function]] - [[Estimating Mean and Variance of Grouped Data]] - [[Exponential Function and Its Integral]] - [[Derivative of the Exponential Function]] - [[Derivative of e_x Using the Limit Definition]] - [[Area Under a Curve]] - [[Improper Integrals]] - [[Definite Integral]] - [[Integration by Parts]] - [[Inflection Point]] - [[Second Derivative Test]] - [[Critical Points]] - **Standard Normal Distribution:** The special case of the normal distribution with $\mu = 0$ and $\sigma = 1$. - **Central Limit Theorem:** Explains why normal distributions arise in many situations. - **Probability Density Functions (PDFs):** General framework for describing continuous random variables. ## Quick Review Questions 1. What are the roles of $\mu$ and $\sigma$ in defining a normal distribution? 2. How do you convert a normally distributed variable $X \sim N(\mu, \sigma^2)$ to a standard normal variable $Z$? *** ![[Normal_Distribution_visualization]]