Normal Probability Distributions: Study Notes

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: $P(a \leq X \leq b)$ is the area under the PDF from $a$ to $b$.

Example: For a die roll (discrete), $P(1 \leq X \leq 3) = P(X=1) + P(X=2) + P(X=3) = \frac{3}{6} = 0.5$. 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: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$

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

• Probability Calculation: $P(c \leq X \leq d) = (d-c) \cdot f(x)$

Example: If response time is uniformly distributed between 2 and 12 seconds, the probability that a call is answered in 5-9 seconds is $P(5 \leq X \leq 9) = \frac{9-5}{12-2} = 0.4$.

Probability Density Functions (PDFs)

A valid PDF must satisfy two conditions: (1) $f(x) \geq 0$ for all $x$, 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 $f(x)$ over its domain; result must be 1.

• Probability: $P(a < X < b) = \int_a^b f(x) dx$

Standard Normal Distribution

The standard normal distribution is a normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$. Probabilities are found using the standard normal (z) table, which gives cumulative probabilities from the left up to a given z-score.

• Z-Score: $z = \frac{x - \mu}{\sigma}$

• Finding Probabilities: Use the z-table to find $P(Z < z)$ or $P(Z > z)$.

• Area to the Right: $P(Z > z) = 1 - P(Z < z)$

• Area Between Two Z-Scores: $P(a < Z < b) = P(Z < b) - P(Z < a)$

Example: $P(Z < -0.64) = 0.2611$ (from z-table), $P(Z > 2.27) = 1 - 0.9884 = 0.0116$.

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 $P(Z < z) = p$.

• Given Area to Right: Find z such that $P(Z > z) = p$; use $P(Z < z) = 1 - p$.

Example: If $P(Z < z) = 0.6331$, 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: $\mu = 0$, $\sigma = 1$

Non-Standard Normal Distributions

For normal distributions with mean $\mu \neq 0$ or standard deviation $\sigma \neq 1$, convert values to z-scores before using the standard normal table or calculator.

• Z-Score Formula: $z = \frac{x - \mu}{\sigma}$

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

Example: Commute times are normal with $\mu = 20$, $\sigma = 5$. Probability of less than 10 minutes: $z = \frac{10-20}{5} = -2$, $P(Z < -2) = 0.023$.

Finding Values from Probabilities (Non-Standard Normal)

To find the value $x$ 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: $x = z \cdot \sigma + \mu$

Example: Heights of women are normal with $\mu = 160$ cm, $\sigma = 7$ cm. Find $x$ such that 5% are shorter than $x$: $z = -1.64$, $x = -1.64 \times 7 + 160 = 148.52$ cm.

Summary Table: Key Formulas and Functions

Concept	Formula/Function	Description
Uniform PDF	$f(x) = \frac{1}{b-a}$	For $a \leq x \leq b$
Probability (Uniform)	$P(c \leq X \leq d) = (d-c) \cdot f(x)$	Area under PDF
Z-Score	$z = \frac{x - \mu}{\sigma}$	Standardizes value
Probability (Standard Normal)	z-table, calculator	Find area under curve
Value from Probability	$x = z \cdot \sigma + \mu$	Reverse standardization

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).