Normal Probability Distributions: Concepts, Properties, and Applications

Normal Probability Distributions

Introduction

Normal probability distributions are fundamental in statistics, describing many natural phenomena and measurement errors. This chapter introduces the standard normal distribution, its properties, and applications, including the empirical rule and calculations involving areas under the curve.

The Standard Normal Distribution

Definition and Properties

• Bell-shaped curve: The graph of the standard normal distribution is symmetric and bell-shaped.

• Mean (μ): The mean is 0.

• Standard deviation (σ): The standard deviation is 1.

The standard normal distribution is denoted as Z ~ N(0, 1).

Probability Density Function (PDF)

• The PDF for the standard normal distribution is:

$f(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}z^2}$ for all $z$ in $(-\infty, \infty)$

• The total area under the curve is 1.

• Probabilities correspond to areas under the curve.

Continuous Distributions

Probability Density Function (PDF)

• $f(x)$ is a continuous function of $x$.

• For a given $X = x$, $f(x)$ is the height of the curve at $x$.

• $f(x)$ is not the probability that $X = x$.

• The probability $P(x_1 \leq X \leq x_2)$ is the area under $f(x)$ between $x_1$ and $x_2$.

• The domain of $f(x)$ is called the support.

• For a continuous random variable, $P(X = x) = 0$.

Uniform Distribution

Definition and Properties

• A random variable $X$ has a continuous uniform distribution on $[a, b]$ if all intervals of the same length within $[a, b]$ are equally likely.

• Notation: $X \sim U(a, b)$

• PDF: $f(x) = \frac{1}{b-a}$ for $x$ in $[a, b]$, $a < b$

• Mean: $\mu = \frac{a+b}{2}$

• Standard deviation: $\sigma = \frac{b-a}{\sqrt{12}}$

Example: Waiting Times for Airport Security

• Suppose waiting times are uniformly distributed between 0 and 5 minutes.

• Probability that a randomly selected passenger waits at least 2 minutes:

$P(X \geq 2) = \text{height} \times \text{width} = 0.2 \times 3 = 0.6$

• Thus, there is a 60% chance a passenger waits at least 2 minutes.

Normal Distribution

Definition and Properties

• A random variable $X$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$:

$X \sim N(\mu, \sigma)$

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{x-\mu}{\sigma}\right)^2}$ for all $x$ in $(-\infty, \infty)$, $\sigma > 0$

• The mean and standard deviation are $\mu$ and $\sigma$, respectively.

Standardization (z-scores)

• Any normal distribution can be converted to the standard normal distribution using:

$z = \frac{x - \mu}{\sigma}$

• This allows the use of standard normal tables or technology for probability calculations.

Applications of the Normal Distribution

Finding Probabilities

• Probabilities correspond to areas under the normal curve.

• Use cumulative distribution tables or software (e.g., Excel's NORM.DIST function).

Example: Bone Density Test

• Bone density z-scores are normally distributed with mean 0 and standard deviation 1.

• Probability that a randomly selected person has a z-score less than 1.27:

$P(Z < 1.27) = 0.8980$

• Interpretation: 89.80% of people have bone density levels below 1.27.

Areas to the Right or Between Values

• To find the area to the right of $z = -1.00$:

$P(Z > -1.00) = 1 - P(Z < -1.00) = 1 - 0.1587 = 0.8413$

• To find the area between $z = -1.00$ and $z = -2.50$:

$P(-2.50 < Z < -1.00) = P(Z < -1.00) - P(Z < -2.50) = 0.1587 - 0.0062 = 0.1525$

• Interpretation: 15.25% of people have bone density readings between -1.00 and -2.50.

The Empirical Rule

Key Intervals and Probabilities

• Approximately 68.27% of data falls within $\mu \pm 1\sigma$

• Approximately 95.45% of data falls within $\mu \pm 2\sigma$

• Approximately 99.73% of data falls within $\mu \pm 3\sigma$

These intervals are independent of the specific values of $\mu$ and $\sigma$.

Practice Problems and Applications

Example: Protein Lengths

• Given $\mu = 530$, $\sigma = 80$, what percent of proteins are between 370 and 690 amino acids?

• Solution: $370$and $690$ are $2\sigma$ below and above the mean, so approximately 95% (Empirical Rule).

Example: Head Circumference for Helmets

• Mean = 22.8 in, $\sigma = 1.1$ in. The middle 98% corresponds to $\mu \pm 2.33\sigma$.

• Range: $22.8 - 2.33 \times 1.1 = 20.24$ in to $22.8 + 2.33 \times 1.1 = 25.36$ in.

Summary Table: Key Properties of Distributions

Distribution	Notation	PDF	Mean	Standard Deviation
Uniform	$X \sim U(a, b)$	$f(x) = \frac{1}{b-a}$ for $x$ in $[a, b]$	$\frac{a+b}{2}$	$\frac{b-a}{\sqrt{12}}$
Normal	$X \sim N(\mu, \sigma)$	$\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{x-\mu}{\sigma}\right)^2}$	$\mu$	$\sigma$
Standard Normal	$Z \sim N(0, 1)$	$\frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}z^2}$	0	1

Key Formulas

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

• Finding $x$ from $z$: $x = \mu + z\sigma$

Summary

• Normal distributions are central to statistical inference and data analysis.

• Probabilities are found as areas under the curve, using tables or technology.

• The empirical rule provides quick estimates for probabilities within 1, 2, or 3 standard deviations of the mean.