The probability that a continuous random variable x falls between two values a and b is defined by the integral of the probability density function, often abbreviated as PDF. The formula is:
P(a < x < b) = ∫ab f(x) dx
Where:
- P(a < x < b) represents the probability that the random variable x falls between the values a and b.
- ∫ab denotes the definite integral from a to b.
- f(x) is the probability density function.
- dx indicates that the integration is with respect to x.
Understanding the Probability Density Function (PDF)
The probability density function, f(x), describes the relative likelihood for a continuous random variable to take on a given value. Unlike a probability mass function (PMF) for discrete variables, the PDF itself does not directly give the probability of a specific value. Instead, the area under the curve of the PDF over a given interval represents the probability of the variable falling within that interval.
Key Properties of a PDF
- Non-negativity: f(x) ≥ 0 for all x. The PDF cannot have negative values.
- Normalization: ∫-∞∞ f(x) dx = 1. The total area under the PDF curve must equal 1, representing the total probability of all possible outcomes.
Example:
Let's say we have a random variable x with a probability density function defined as f(x) = 2x for 0 ≤ x ≤ 1 and f(x) = 0 otherwise. To find the probability that x falls between 0.2 and 0.6, we would calculate the definite integral:
P(0.2 < x < 0.6) = ∫0.20.6 2x dx
Evaluating this integral gives:
P(0.2 < x < 0.6) = [x2]0.20.6 = (0.6)2 - (0.2)2 = 0.36 - 0.04 = 0.32
Therefore, the probability that x falls between 0.2 and 0.6 is 0.32.
Importance of the PDF
The PDF is a fundamental concept in probability and statistics, allowing us to:
- Calculate probabilities for continuous random variables.
- Model and analyze real-world phenomena that exhibit continuous variation.
- Make predictions and inferences based on observed data.
