Normal Probability Distributions

Introduction to Continuous Random Variables

Continuous random variables (CRVs) can take any value within a given range and are not restricted to isolated points, unlike discrete random variables. Probabilities for CRVs are determined by calculating the area under a probability density function (PDF) over a specified interval.

• Discrete Random Variable: Can only take specific, separate values (e.g., number of die rolls).

• Continuous Random Variable: Can take any value within an interval (e.g., time, distance).

• Probability for CRV: is the area under the PDF from to .

Example: For a die roll (discrete), . For a continuous variable, probability is found by integrating the PDF over the interval.

Uniform Distribution

The uniform distribution is a type of continuous probability distribution where every value within a certain interval is equally likely. The PDF is constant over the interval.

• Probability Density Function: for

• Total Area: The total area under the PDF is always 1.

• Probability Calculation:

Example: If response time is uniformly distributed between 2 and 12 seconds, the probability that a call is answered in 5-9 seconds is .

Probability Density Functions (PDFs)

A valid PDF must satisfy two conditions: (1) for all , and (2) the total area under the curve is 1. Probabilities are found by calculating the area under the curve for the desired interval.

• Check for Valid PDF: Integrate over its domain; result must be 1.

• Probability:

Standard Normal Distribution

The standard normal distribution is a normal distribution with mean and standard deviation . Probabilities are found using the standard normal (z) table, which gives cumulative probabilities from the left up to a given z-score.

• Z-Score:

• Finding Probabilities: Use the z-table to find or .

• Area to the Right:

• Area Between Two Z-Scores:

Example: (from z-table), .

Finding Z-Scores from Probabilities

To find the z-score corresponding to a given probability (area), use the z-table in reverse. If the area is to the right, subtract from 1 to get the area to the left, then look up the z-score.

• Given Area to Left: Find z such that .

• Given Area to Right: Find z such that ; use .

Example: If , z-score is approximately 0.34 (from z-table).

Using the TI-84 Calculator for Normal Probabilities

The TI-84 calculator can quickly compute probabilities and z-scores for normal distributions using built-in functions.

• Finding Probability from Z-Score: Use normalcdf or ShadeNorm functions.

• Finding Z-Score from Probability: Use invNorm function.

• Standard Normal Parameters: ,

Non-Standard Normal Distributions

For normal distributions with mean or standard deviation , convert values to z-scores before using the standard normal table or calculator.

• Z-Score Formula:

• Finding Probabilities: Convert to , then use the z-table or calculator.

Example: Commute times are normal with , . Probability of less than 10 minutes: , .

Finding Values from Probabilities (Non-Standard Normal)

To find the value corresponding to a given probability, first find the z-score from the probability, then solve for $x$ using the mean and standard deviation.

• Value from Probability:

Example: Heights of women are normal with cm, cm. Find such that 5% are shorter than $x$: , cm.

Summary Table: Key Formulas and Functions

Additional info: These notes cover the essentials of continuous random variables, the uniform and normal distributions, and practical calculation methods using tables and calculators, as outlined in a typical college statistics course (Ch. 6 - Normal Probability Distributions).