Probability Concepts and Standard Normal Distribution
Probability of Events
Calculating Probability
- Define the experiment and observation process.
- List the sample space (SP).
- Assign probabilities to each outcome in the SP.
- Identify the outcomes within the event of interest.
- Sum the probabilities of those outcomes.
Set Operations
- Union (A u B): Occurs if either A, B, or both occur. Includes all outcomes in A or B or both.
- Intersection (A n B): Occurs if both A and B occur. Includes outcomes common to both A and B.
- Complement (A^c): Occurs if A does not occur. Includes all outcomes not in A.
Rule of Complements:
P(A) + P(A^c) = 1
Probability Rules
- Mutually Exclusive Events: P(A u B) = P(A) + P(B)
- Non-Mutually Exclusive Events: P(A u B) = P(A) + P(B) – P(A n B)
Random Variables
- Discrete: Can take a countable number of values.
- Continuous: Can take any value within a range.
Probability Distributions
A probability distribution describes the probability of each possible value of a random variable.
Discrete Random Variable:
Represented by a table, graph, or formula.
Continuous Random Variable:
Represented by a smooth curve (probability density function).
Standard Normal Distribution
A normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. Denoted by Z.
Z-Scores
A Z-score measures how many standard deviations a data point is from the mean.
Z-Table
Provides cumulative probabilities (area under the curve) for Z-scores.
Using the Z-Table
- P(Z < z): Find the cumulative probability up to z.
- P(Z > z): Subtract the table value from 1 to get the area to the right of z.
- P(|Z| < z): Find the area between -z and z (double the table value for z).
- P(|Z| > z): Find the area in both tails beyond ±z.
Finding Z-Scores for Given Probabilities
Use the Z-table in reverse to find the Z-score corresponding to a given cumulative probability.
Probabilities Between Two Z-Scores
Subtract the cumulative probability of the smaller Z-score from the cumulative probability of the larger Z-score.
Probabilities Within a Range
For P(|Z| ≤ z), double the cumulative probability for the positive Z-score.
Sample Statistics and Parameters
- Sample Statistic: A numerical measure calculated from a sample.
- Sampling Distribution: The probability distribution of a sample statistic.
- Parameter: A numerical measure describing a population.