Chapter 6: Normal Probability Distributions – Study Notes

Normal Probability Distributions

Introduction to Probability Distributions

Probability distributions describe how the values of a random variable are distributed. They are fundamental in statistics for modeling and analyzing random phenomena. There are two main types: discrete and continuous probability distributions.

• Random Variable: A variable whose value is determined by chance, for each outcome of a procedure.

• Probability Distribution: Specifies the probability for each value of the random variable.

• Discrete Random Variable: Has a finite or countable number of possible values (e.g., 0, 1, 2, ...).

• Continuous Random Variable: Has infinitely many values, often associated with measurements on a continuous scale.

Key Symbols: For discrete random variables, probabilities are denoted by P(x). For continuous random variables, probabilities are denoted by P(a < x < b).

Uniform Probability Distribution

A uniform distribution is a type of continuous probability distribution where all outcomes in a given range are equally likely. The probability density function (pdf) for a uniform random variable X over the interval [a, b] is:

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

• Mean: $\mu = \frac{a+b}{2}$

• Variance: $\sigma^2 = \frac{(b-a)^2}{12}$

Example: If the length of a class is uniformly distributed between 50.3 and 50.9 minutes, the probability that a class lasts between 50.3 and 50.9 minutes is $P(50.3 < x < 50.9) = \frac{50.9-50.3}{50.9-50.3} = 1$.

The Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. It is one of the most important distributions in statistics, modeling many natural phenomena.

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

• Parameters: $\mu$ (mean), $\sigma$ (standard deviation)

• Properties: Symmetric about the mean, total area under the curve is 1, extends infinitely in both directions.

Example: Heights of adult men and women are often modeled using the normal distribution.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$. The random variable is denoted by z.

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

• Standard Normal Variable: $z = \frac{x - \mu}{\sigma}$

• Use: Probabilities for any normal distribution can be found by converting to the standard normal variable and using standard normal tables.

Example: To find $P(x < a)$ for a normal variable, convert $x$ to $z$ and use the standard normal table.

Using the Standard Normal Table

The standard normal table (Table A-2) provides cumulative probabilities for values of z. The table gives $P(z < a)$, the area under the curve to the left of z.

• Example Calculation: $P(z < 1.33) = 0.9082$

• Interval Probability: $P(0.17 < z < 2.09) = P(z < 2.09) - P(z < 0.17) = 0.9814 - 0.5675 = 0.4139$

Note: The distribution is symmetric about the mean. For negative z values, use symmetry: $P(z < -a) = 1 - P(z < a)$.

Basic Properties of the Standard Normal Curve

• Property 1: Total area under the curve is 1.

• Property 2: The curve extends infinitely in both directions, never touching the horizontal axis.

• Property 3: The curve is symmetric about the mean ($z = 0$).

• Property 4: Almost all the area lies between $z = -3$ and $z = 3$.

The Empirical Rule

The Empirical Rule describes the percentage of data within certain intervals in a normal distribution:

• About 68% of data falls within 1 standard deviation ($\mu \pm \sigma$)

• About 95% within 2 standard deviations ($\mu \pm 2\sigma$)

• About 99.7% within 3 standard deviations ($\mu \pm 3\sigma$)

HTML Table: Standard Normal Table (Excerpt)

The standard normal table provides cumulative probabilities for z values. Below is an excerpt:

z	P(z < z)
0.00	0.5000
0.50	0.6915
1.00	0.8413
1.33	0.9082
2.00	0.9772

Additional info: Table values are cumulative from the left; for negative z, use symmetry.

Summary

• The normal distribution is central to statistics, modeling many real-world phenomena.

• Probabilities are found using the standard normal distribution and tables.

• The Empirical Rule helps estimate the spread of data in a normal distribution.