STAT1150 - Unit 2: Probability, Density Curves & Normal Distributions Cheat Sheet

Probability quantifies the likelihood of outcomes in random phenomena, with the long-run behavior of repeated trials revealing predictable patterns. Density curves, particularly the normal distribution, provide a smooth mathematical model for these distributions, where the area under the curve represents probability.

Core Principles

Probability is the long-run proportion of times an outcome occurs in an infinite series of trials.
A probability distribution mathematically models random behavior by listing possible outcomes and their associated probabilities.
Density curves are smooth models of distributions where the total area under the curve is 1, and the area between two points represents the probability of observations falling within that range.
The normal distribution is a symmetric, bell-shaped density curve defined by its mean (μ) and standard deviation (σ), where μ indicates the center and σ indicates the spread.
The 68-95-99.7 rule states that for any normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
The z-score standardizes a value from any normal distribution to the standard normal distribution (μ=0, σ=1), allowing for probability calculations using z-tables or software.

Action Steps

To find a probability for a normal distribution: 1. Standardize the value(s) to z-scores using $z = \frac{x - \mu}{\sigma}$. 2. Use the z-table or statistical software to find the corresponding probability (area under the standard normal curve).
To find a value (x) given a probability for a normal distribution: 1. Find the z-score corresponding to the given probability using the z-table (working backward). 2. Unstandardize the z-score to find x using $x = \mu + z\sigma$.

Formulas

Z-score: $z = \frac{x - \mu}{\sigma}$
Solving for σ: $\\sigma = \frac{x - \mu}{z}$
Solving for μ: $\\mu = x - z\sigma$
Area of a circle: $Area = \pi r^2$

Key Terms

Chance experiment: An activity or process where the outcome is uncertain.
Outcome: A single possible result of a chance experiment.
Event: A set of one or more outcomes from a chance experiment.
Sample space (S): The set of all possible outcomes of a chance experiment.
Probability: The proportion of times an outcome would occur in an infinitely long series of trials.
Probability distribution: A mathematical model listing all possible outcomes and their probabilities.
Density curve: A smooth curve representing a distribution where the total area is 1 and area under intervals represents probability.
Normal distribution: A symmetric, bell-shaped density curve defined by mean (μ) and standard deviation (σ).
Standard normal distribution: A normal distribution with a mean of 0 and a standard deviation of 1 (Z ~ N(0,1)).
Z-score: A measure of how many standard deviations an observation is from the mean.
Parameter: A numerical measurement describing an entire population (e.g., population mean μ).
Statistic: A numerical measurement computed from a sample, used to estimate a parameter (e.g., sample mean x̄).
Random variable: A numerical description of the outcome of a statistical experiment.
Percentile: The value below which a given percentage of observations in a group of observations fall.

Real World Examples

Coin tossing: In the short run (e.g., 10 trials), proportions of heads vary. In the long run (e.g., 1000 trials), the proportion approaches the theoretical probability of 0.5.
Midterm marks in a biology class: With a mean of 65 and standard deviation of 10, 68% of students scored between 55 and 75, 95% between 45 and 85, and 99.7% between 35 and 95.
Pulse rates of adult females: A mean of 74 bpm and standard deviation of 12 bpm allows calculation of proportions (e.g., 92.22% have rates above 57 bpm) and percentiles (e.g., the lowest 10% are below 58.64 bpm).