Study Notes
Probability distributions describe how probabilities are distributed over the values of a random variable. They can be discrete or continuous, with different functions to represent them.
- Random Variable — a variable whose value is the outcome of a random phenomenon.
Example: Rolling a die, where the outcome is a number from 1 to 6. - Discrete Random Variable — takes on a countable number of distinct values.
Example: Number of heads in 10 coin flips. - Continuous Random Variable — can take any value within a given range.
Example: The height of students in a class. - Probability Distribution Function (PDF) — gives the probability that a discrete random variable is exactly equal to some value.
Example: P(X=x) for a die roll. - Cumulative Distribution Function (CDF) — gives the probability that a random variable is less than or equal to a certain value.
Example: F(x) = P(X ≤ x). - Binomial Distribution — a probability distribution of a binomial random variable, which counts the number of successes in a fixed number of trials.
Example: Flipping a coin 10 times and counting the number of heads. - Normal Distribution — a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence.
Example: Heights of people in a population.
Exam Tips
Key Definitions to Remember
- Random Variable
- Discrete Random Variable
- Continuous Random Variable
- Probability Distribution Function (PDF)
- Cumulative Distribution Function (CDF)
- Binomial Distribution
- Normal Distribution
Common Confusions
- Mixing up discrete and continuous random variables
- Confusing PDF with CDF
- Misunderstanding the conditions for a binomial distribution
Typical Exam Questions
- What is the probability that a discrete random variable equals a specific value? Use the PDF to find P(X=x).
- How do you calculate the expected value of a random variable? Use E(X) = ΣxP(X=x).
- What is the probability of getting at most k successes in a binomial distribution? Use the CDF for binomial distribution.
What Examiners Usually Test
- Understanding and application of probability distribution functions
- Ability to calculate probabilities using PDFs and CDFs
- Application of binomial and normal distributions in real-world contexts