Continuous Probability Distributions and the Normal Distribution

Definition of Discrete Variables (Chapter 5 Recap): - Discrete variables deal with countable outcomes. - Example: Outcomes like 1, 2, and 3 are distinct, individual points on a number line.
Definition of Continuous Variables (Chapter 6): - Continuous variables deal with uncountable outcomes. - Instead of distinct points, the outcome is a range (e.g., any real number between 1 and 3). - Example: A value could be $1.00000001$ , $1.00000002$ , or $2.9999999$ . - These variables typically arise from a measuring process rather than a counting process.
Probability Calculation for Exact Values: - Discrete Side: It is possible to calculate the probability for an exact value, such as $P(X = 5)$ . - Continuous Side: The probability of an exact value is always zero ( $P(X = 5) = 0$ ). - Reasoning: Because there are an infinite number of possible outcomes (e.g., $5.1$ , $5.2$ , $4.9$ , $4.99$ ), the chance of landing on one specific, exact point is so small it is mathematically considered zero.
Probability Over Intervals: - Because exact points cannot be calculated, probabilities for continuous variables are calculated over intervals. - Example: Instead of finding $P(X = 5)$ , we calculate P(4.5 < X < 5.5). This serves as a close approximation for the probability of the value being roughly five.

Probability Mass Function (PMF): - Used for discrete distributions (Chapter 5). - Graphing: The y-axis represents the probability of $x$ ( $P(X)$ ) and the x-axis represents the outcome $x$ . - Representation: Probabilities are shown as the height of a line or bar at each distinct integer.
Probability Density Function (PDF): - Used for continuous distributions (Chapter 6). - Graphing: The x-axis represents $x$ , but the y-axis represents $f(x)$ , which is a function that gives probability density. - Representation: It is a smooth, continuous function. - Crucial Metric: Probability is represented as the area under the curve between two bounds ( $a$ and $b$ ), rather than the height at a single point.
Core Laws of Probability: - Discrete Rule: The sum of all individual probabilities must equal one: $\sum P(x) = 1$ . - Continuous Rule: The total area under the density curve must equal one, as this representing the entire sample space. - Integration: To find the area under the curve (the probability), one must integrate the function. For a theoretical range, integrating from negative infinity to positive infinity equals one: $\int_{-\infty}^{\infty} f(x)\,dx = 1$ .

Definition: The Normal Distribution is the most common continuous probability distribution.
Rationale for the Name "Normal": - It has a very predictable shape. - It is highly prevalent in real-life data collection; gathered data often naturally takes this distribution shape when plotted.
Characterizing Parameters: The distribution is uniquely defined by two metrics: - Mean ( $\mu$ ): This represents the center of the distribution. - Standard Deviation ( $\sigma$ ): This represents the spread, or how much data points deviate from the mean on average.
Notation: The notation for a normal distribution is $X \sim N(\mu, \sigma)$ or $X \sim N(\mu, \sigma^2)$ . If you know the variance, you can find the standard deviation by taking the square root; if you have the standard deviation, you square it to find the variance.

Mathematical Form: The function $f(x)$ that creates the specific bell-shaped curve is defined as: - $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} (\frac{x-\mu}{\sigma})^2}$
Constants within the Function: - $\pi$ : This is the mathematical constant approximately equal to $3.14$ . - $e$ : The base of the natural logarithm.
Calculus Context: To find the probability that $x$ lies between bounds $a$ and $b$ , one would technically integrate the function: - P(a < X < b) = \int_a^b \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} (\frac{x-\mu}{\sigma})^2} \,dx - Note: This integral has no closed-form solution, meaning it cannot be solved with standard algebraic integration. Historically, it required brute-force computation or approximation.

Definition: A specific case of the normal distribution where the mean is zero and the standard deviation is one ( $\mu = 0, \sigma = 1$ ).
Standardization Process: Converting an $x$ value to a $z$ value scales any normal distribution so it can be compared to the standard normal table.
Conversion Formula: - $Z = \frac{x - \mu}{\sigma}$
Purpose: Tables and calculators are generally programmed based on the Standard Normal Distribution. By converting $x$ to $z$ , you can look up probabilities for any distribution regardless of its original mean or standard deviation.

Convert $X$ to $Z$ : Use the formula $Z = \frac{x - \mu}{\sigma}$ . This transforms the statement from the $X$ scale to the $Z$ scale (e.g., P(X < 18.6) becomes P(Z < 0.12)).
Look up the Probability: Use a Standard Normal Table ( $Z$ -table) or a calculator. - Lower Tail Probability: Most tools default to providing the probability that $Z$ is less than or equal to a value ( $P(Z \leq z)$ ).

Reason for Requirement: The HP calculator is necessary for this module because it stores the standard normal table results internally.
Step-by-Step Calculator Process: - Value Input: Enter the calculated $z$ value (e.g., $0.12$ ). - Function: Use the "Blue Shift" followed by the "3" button. Above the 3 key is the notation $Z \rightarrow P$ , which converts a $z$ value into a probability. - Adjusting Decimals: To see more precision, press "Orange Shift", then "=" (Display), then a number (e.g., 4 or 5) to set the decimal places.
Table Lookup Alternative: In a $z$ -table, you find the first part of the value (e.g., $0.1$ ) in the first column and the second decimal (e.g., $0.02$ ) in the top row. The intersection provides the probability (e.g., $0.5478$ ).

Case Study: File Download Times: - Given: $x$ is the download time in seconds (continuous). - Parameters: $X \sim N(\mu=18, \sigma=5)$ . - Scenario A: Finding P(X < 18.6): - Convert: $Z = \frac{18.6 - 18}{5} = 0.12$ . - Probability: P(Z < 0.12) = 0.5478. - Scenario B: Upper Tail Probabilities (P(X > 18.6)): - Because the total area is 1, and calculators/tables typically give the lower tail (left side), use the complement rule. - Formula: P(X > a) = 1 - P(X < a). - Calculation: 1 - P(Z < 0.12) = 1 - 0.5478 = 0.4522.
The Continuity Property (Inequalities): - In continuous distributions (Chapter 6), it does not matter if a sign is "strictly less than" (<) or "less than or equal to" ( $\leq$ ). - P(X < 18.6) = P(X \leq 18.6). - Explanation: The difference between the two is a single point ( $18.6$ ), and the probability of a single point in a continuous distribution is zero. Therefore, there is no meaningful difference. - Warning: This logic cannot be applied to discrete distributions (Chapter 5), where the difference between integers is significant.