Statistics and Probability - Continuous Probability Distributions (Normal Distribution)

The Normal Distribution is a continuous probability distribution defined by the notation $X \sim N(\mu, \sigma^2)$, where $\mu$ is the mean and $\sigma^2$ is the variance. Visually, it is represented by a 'bell-shaped' curve that is perfectly symmetrical about the vertical line $x = \mu$. The peak of the curve occurs at the mean, which is also the mode and the median for this distribution.

The total area under the normal curve is exactly $1$, representing a total probability of $100\%$. Because of its symmetry, the area to the left of the mean $P(X < \mu)$ is $0.5$, and the area to the right $P(X > \mu)$ is also $0.5$. On a graph, this means the curve is divided into two identical halves by the mean line.

The parameters $\mu$ and $\sigma$ control the shape and position of the distribution. Changing $\mu$ shifts the entire bell curve horizontally along the x-axis without changing its shape. Changing the standard deviation $\sigma$ affects the 'spread': a small $\sigma$ results in a tall, thin curve clustered tightly around the mean, while a large $\sigma$ results in a shorter, wider, and flatter curve.

The Empirical Rule (or the 68-95-99.7 rule) describes the spread of data: approximately $68.27\%$ of the data lies within one standard deviation of the mean ($[\mu - \sigma, \mu + \sigma]$), $95.45\%$ lies within two standard deviations ($[\mu - 2\sigma, \mu + 2\sigma]$), and $99.73\%$ lies within three standard deviations. Visually, the 'tails' of the curve approach the x-axis but never actually touch it (asymptotic behavior).

The Standard Normal Distribution is a specific normal distribution where $\mu = 0$ and $\sigma = 1$, denoted as $Z \sim N(0, 1)$. Any normal variable $X$ can be transformed into a $Z$-score, which represents the number of standard deviations an observation is from the mean. This is visually useful for comparing different datasets on a standardized scale.

Probability for a continuous distribution is defined as the area under the curve between two points. For the Normal Distribution, the probability $P(a < X < b)$ is the integral of the probability density function from $a$ to $b$. Crucially, for any continuous distribution, the probability of the variable equaling an exact value is zero, $P(X = k) = 0$.

The Inverse Normal Distribution is the process of finding a boundary value $x$ given a known area (probability). For example, if you are given that the bottom $10\%$ of students failed a test, you use the inverse normal function on your GDC to find the score $x$ such that $P(X < x) = 0.10$. Visually, this corresponds to finding the x-coordinate that cuts off a specific area in the left tail of the curve.