Overview
This lecture introduces continuous random variables and their probability distributions, focusing on how probabilities are calculated using probability density functions and areas under curves.
Continuous Random Variables
- Continuous random variables can take any value within an interval, including infinitely many possible values.
- Examples include height, time to failure, or any measurement along a continuum.
- The variable can take on values like 3.1, 3.12, or any number in the range.
Probability Density Functions (PDF)
- Continuous random variables are modeled using a probability density function (f(x)), a curve describing likelihoods.
- The PDF gives the height of the curve at each value of x.
- For continuous variables, probability is represented by the area under the curve, not the height.
Calculating Probabilities
- Probability that a continuous random variable falls between values a and b is the area under the curve from a to b.
- The probability of the variable being exactly any specific value (e.g., P(X = a)) is always zero.
- It makes sense only to discuss probabilities for intervals, not single points.
Properties of PDFs
- f(x) must be at least zero for all x (f(x) ≥ 0).
- The total area under the PDF curve is exactly one.
- f(x) can have values greater than one, but total area stays one.
Common Continuous Distributions
- The normal distribution is a key example, with its characteristic bell-shaped curve.
- The uniform distribution has a constant f(x) over its interval.
- The exponential distribution is used for modeling decay and similar processes.
Calculating Probabilities & Percentiles
- Probabilities and percentiles are computed by integrating the PDF over desired intervals.
- Statistical software or tables are typically used to compute these integrals in practice.
Key Terms & Definitions
- Continuous random variable — A variable that can take any value within an interval.
- Probability Density Function (PDF) — A function f(x) giving the likelihood of a random variable at value x.
- Area under the curve — Represents the probability a variable falls within a certain range.
- Normal distribution — A symmetric, bell-shaped continuous distribution.
- Uniform distribution — A distribution where all values in an interval are equally likely.
- Exponential distribution — A distribution often modeling time to an event (e.g., decay).
Action Items / Next Steps
- Review examples of continuous probability distributions in your textbook.
- Practice calculating probabilities as areas under curves using PDFs.
- Familiarize yourself with using statistical tables or software for integration.