Chapter 6

Chapter 6: Modelling Random Events – The Normal and Binomial Models

Learning Objectives

• Define probability distributions of random variables.

• Distinguish between discrete and continuous random variables.

• Explain and construct probability distributions for both types.

• Understand and apply the normal probability model and the binomial probability model.

Section 6.1: Probability Distributions

Probability Distributions

A probability distribution describes the probabilities of all possible outcomes of a random variable. It provides a model for the likelihood of different results in a random experiment.

• Random Variable: A variable whose value is a numerical outcome of a random phenomenon.

• Probability Distribution: Assigns a probability to each possible value of the random variable.

Example: Rolling a die. The probability distribution for the number showing up is:

Value	1	2	3	4	5	6
Probability	1/6	1/6	1/6	1/6	1/6	1/6

Types of Random Variables

• Discrete Random Variables: Take on countable values (e.g., number of heads in coin tosses).

• Continuous Random Variables: Take on any value in an interval (e.g., height, weight).

Example (Discrete): Number of siblings a person has. Example (Continuous): Height of students in a class.

Probability Distributions for Discrete Random Variables

• List all possible values and their probabilities.

• Probabilities must satisfy: $0 \leq P(x) \leq 1$ for all $x$ and $\sum P(x) = 1$.

Example: Rolling a fair die. Each outcome (1 to 6) has probability $1/6$.

Probability Distributions for Continuous Random Variables

For continuous random variables, probabilities are assigned to intervals, not exact values. The probability that $X$ falls within an interval $[a, b]$ is the area under the curve of the probability density function (pdf) between $a$ and $b$.

• The total area under the pdf curve is 1.

Example: The probability that a randomly chosen person is between 160 cm and 170 cm tall is the area under the height distribution curve between those values.

Finding Probabilities as Areas

• For discrete variables: Sum probabilities for the desired outcomes.

• For continuous variables: Calculate the area under the pdf curve for the interval of interest.

Example: For a uniform distribution between 0 and 1, the probability that $X$ is between 0.4 and 0.7 is $0.7 - 0.4 = 0.3$.

Section 6.2: The Normal Model

The Normal Model

The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. It is used to model many natural phenomena, such as heights, test scores, and measurement errors.

• Characterized by its mean ($\mu$) and standard deviation ($\sigma$).

• Notation: $N(\mu, \sigma)$.

Properties of the Normal Distribution

• Symmetric about the mean.

• Mean, median, and mode are equal.

• Described completely by $\mu$ and $\sigma$.

• Empirical Rule: About 68% of data within 1$\sigma$, 95% within 2$\sigma$, 99.7% within 3$\sigma$ of the mean.

Standard Normal Distribution

• A special case with $\mu = 0$ and $\sigma = 1$.

• Standardized values are called z-scores:

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

Finding Normal Probabilities

• Probabilities correspond to areas under the normal curve.

• Use z-tables or technology to find areas/probabilities for given z-scores.

Example: Find $P(z < 1.04)$. From the z-table, $P(z < 1.04) = 0.8508$.

Using Technology to Find Areas and z-values

• Statistical calculators and software can compute normal probabilities and percentiles.

• Commands such as normalcdf and invNorm are used on calculators.

Example: To find the area to the left of $z = 1.5$, use normalcdf(-1000, 1.5, 0, 1).

Finding Measurements from Percentiles (Inverse Normal)

• Given a percentile, find the corresponding value $x$ using the inverse normal function.

$x = \mu + z\sigma$

Example: The 90th percentile of a normal distribution with $\mu = 100$ and $\sigma = 15$ is $x = 100 + (1.28)(15) = 119.2$.

Section 6.3: The Binomial Model

Binomial Probability Model

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Conditions:

  • Fixed number of trials ($n$)

  • Each trial is independent

  • Each trial has two outcomes (success/failure)

  • Probability of success ($p$) is constant

Probability of exactly $k$ successes in $n$ trials:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

where $\binom{n}{k}$ is the binomial coefficient.

Shape of the Binomial Distribution

• The shape depends on $n$ and $p$.

• As $n$ increases, the distribution becomes more symmetric, especially if $p$ is near 0.5.

Mean and Standard Deviation of the Binomial Distribution

• Mean: $\mu = np$

• Standard deviation: $\sigma = \sqrt{np(1-p)}$

Approximating the Binomial with the Normal Distribution

• The binomial distribution can be approximated by the normal distribution when $np \geq 15$ and $n(1-p) \geq 15$.

• This is useful for large $n$ where direct calculation is complex.

Examples and Applications

• Finding binomial probabilities using tables, formulas, or technology.

• Interpreting results in context (e.g., probability of a certain number of successes in a survey).

Comparison of Distributions

Distribution	Type of Variable	Parameters	Example
Discrete (Binomial)	Discrete	$n$, $p$	Number of heads in 10 coin tosses
Continuous (Normal)	Continuous	$\mu$, $\sigma$	Heights of students

Key Formulas

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

• Binomial probability: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Mean of binomial: $\mu = np$

• Standard deviation of binomial: $\sigma = \sqrt{np(1-p)}$

Summary Table: Properties of Distributions

Property	Normal Distribution	Binomial Distribution
Shape	Symmetric, bell-shaped	Symmetric or skewed (depends on $p$)
Parameters	$\mu$, $\sigma$	$n$, $p$
Variable Type	Continuous	Discrete
Application	Heights, test scores	Number of successes in trials

Additional info: These notes are based on lecture slides and include expanded explanations, definitions, and examples for clarity and completeness.