Calculations with Normal Distributions

Probability as Area: In a normal distribution $X \sim N(\mu, \sigma^2)$, the probability $P(a < X < b)$ is represented by the area under the probability density curve between the vertical lines $x = a$ and $x = b$.

Continuous Nature: Because the distribution is continuous, the probability of the variable taking an exact single value is always zero, expressed as $P(X = k) = 0$.

Inequality Equivalence: Due to the zero probability of single points, strict and weak inequalities are interchangeable in calculations: $P(X \le k) = P(X < k)$.

Normal Cumulative Distribution (NCD): This function calculates the probability that a variable falls within a specific range $[a, b]$ by integrating the density function between those bounds.

Handling Infinite Tails: To calculate $P(X > a)$, use a sufficiently large upper bound (e.g., $10^{99}$); for $P(X < b)$, use a sufficiently small lower bound (e.g., $-10^{99}$).

Input Parameters: Most calculation tools require four inputs: the lower bound, the upper bound, the standard deviation ($\sigma$), and the mean ($\mu$).

Finding Values from Area: The inverse normal function is used when the probability (area) $p$ is known, and you need to find the corresponding value $a$ such that $P(X < a) = p$.

Tail Direction: Calculators often default to the 'left tail' (area to the left of $a$); if given a 'right tail' probability $P(X > a) = p$, you must use the complement $P(X < a) = 1 - p$ for standard tools.

Logical Verification: If $P(X < a) > 0.5$, then $a$ must be greater than the mean $\mu$; if $P(X < a) < 0.5$, then $a$ must be less than $\mu$.

The Z-Score: Any normal distribution $X \sim N(\mu, \sigma^2)$ can be mapped to the standard normal distribution $Z \sim N(0, 1)$ using the formula $z = \frac{x - \mu}{\sigma}$.

Purpose of Standardization: This transformation allows for the comparison of different datasets and is essential for finding unknown means or standard deviations when only probabilities are provided.

Symmetry Properties: In the $Z$-distribution, $P(Z < -z) = P(Z > z)$ due to symmetry about the mean of zero.

Always Sketch: Drawing a quick bell curve and shading the relevant area prevents 'tail errors' and provides a visual sanity check for your answer.

Parameter Solving: If both $\mu$ and $\sigma$ are unknown, you will typically be given two probabilities. Use the $z$-score formula to create two simultaneous equations: $x_1 = \mu + z_1\sigma$ and $x_2 = \mu + z_2\sigma$.

Precision: When calculating $z$-scores for further use, keep at least 4 or 5 decimal places to avoid rounding errors in the final answer.

Sanity Check: Ensure your probability is between 0 and 1. If a value $x$ is many standard deviations from the mean, the probability of being beyond it should be very small.