In this section, the calculations involving a normally distributed variable, denoted as $X \sim N(\mu, \sigma^2)$, where $\mu$ is the mean and $\sigma^2$ is the variance, will be explored in-depth. This exploration includes calculating probabilities, deriving relationships between variables, and the standardization process.

Basics of Normal Distribution

• Normal distribution is a bell-shaped curve reflecting a continuous probability distribution.
• Symmetry and Mean: The curve is symmetric around the mean, showing that values near the mean are more common.
• Standard Deviation $(\sigma)$: Determines the distribution's spread. A larger $\sigma$ means a wider spread.
• Real-World Examples: Common in heights, test scores, and measurement errors.

Probability Calculations in Normal Distribution

• Calculating Probabilities: Focuses on the area under the curve.
• Standard Normal Distribution Table: Shows the probability of a standard normal variable $(Z)$ being within a range. It lists probabilities for values less than a given $Z$-score.

Example Problems

Example 1: Finding Probability (P(X > 12))

• Standardize the Variable: Convert $X$ to $Z$ using . $Z = \frac{X - \mu}{\sigma}$ . For $X = 12, Z = 1$.
• Calculate Probability: Find P(Z > 1) \approx 0.1587. This means P(X > 12) = 0.1587, or a 15.87% chance $X$ will be greater than 12.

Example 2: Deriving Value of $(x_1)$

• Find the $Z$-Score: For P(X < x_1) = 0.8413, $Z$-score is about 1.00.
• Calculate $x_1$: Using $x_1 = \mu + Z\sigma$, find $x_1 = 23$. There's an 84.13% chance $X$ will be less than 23.

Example 3: Standardizing a Normal Variable

• Calculate the $Z$-Score: For $X = 35$ in $N(30, 16)$, $Z = 1.25$.
• Graphical Representation: Shows $X = 35$ is $1.25$ standard deviations above the mean of 30.