Section 7.1: Properties of the Normal Distribution – Study Notes

Properties of the Normal Distribution

Objective 1: Use the Uniform Probability Distribution

The uniform probability distribution is a type of continuous probability distribution where all outcomes in a given interval are equally likely. It is often used to model scenarios where the probability of occurrence is constant across the interval.

• Definition: A continuous random variable X has a uniform distribution on the interval [a, b] if its probability density function (pdf) is constant for all x in [a, b].

• Probability Density Function: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$

• Example: If a package is scheduled to arrive between 10 AM and 11 AM, and the arrival time is equally likely at any moment in that interval, the probability that it arrives between 10:15 AM and 10:30 AM is: $P(10.25 \leq X \leq 10.5) = \frac{10.5 - 10.25}{11 - 10} = 0.25$

• Height of the pdf: The height is $\frac{1}{b-a}$, ensuring the total area under the curve is 1.

• Area under the curve: Represents probability for intervals of X.

Objective 2: Graph a Normal Curve

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$).

• Definition: The normal distribution models many natural phenomena and is fundamental in statistics.

• Probability Density Function: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

• Graph Features:

  • The peak of the curve occurs at $x = \mu$.

  • Inflection points are at $x = \mu \pm \sigma$.

  • The curve is symmetric about the mean.

• Normality: A variable is normally distributed if its histogram approximates the bell-shaped curve.

Objective 3: State the Properties of the Normal Curve

The normal curve has several important properties that make it useful for statistical analysis.

• 1. The curve is symmetric about the mean ($\mu$).

• 2. The mean, median, and mode are all equal and located at the center.

• 3. The total area under the curve is 1.

• 4. The curve approaches, but never touches, the x-axis as it extends farther from the mean.

• 5. The inflection points occur at $\mu \pm \sigma$.

• 6. The area under the curve within one standard deviation of the mean is approximately 68%.

• 7. The area within two and three standard deviations are approximately 95% and 99.7%, respectively.

Objective 4: Explain the Role of Area in the Normal Density Function

In the normal distribution, the area under the curve between two values represents the probability that the random variable falls within that interval. This is a key concept for interpreting probabilities and percentiles in normally distributed data.

• Probability Interpretation: The area under the curve between $x_1$ and $x_2$ is $P(x_1 < X < x_2)$.

• Standard Normal Distribution: When $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution.

• Example: If cholesterol levels are normally distributed with $\mu = 214$ and $\sigma = 51$, the area to the right of $x = 200$ represents the probability that a randomly selected individual has cholesterol above 200.

Example Table: Properties of the Normal Curve

Property	Description
Symmetry	Curve is symmetric about the mean
Mean = Median = Mode	All located at the center
Total Area	Area under curve is 1
Inflection Points	At $\mu \pm \sigma$
Empirical Rule	68% within 1$\sigma$, 95% within 2$\sigma$, 99.7% within 3$\sigma$

Additional info: The notes include questions and examples to reinforce understanding of the normal and uniform distributions, their properties, and the interpretation of area under the curve as probability.