Definition
A probability density function (PDF) is a mathematical function that describes the likelihood of a random variable to take on a particular value. The PDF is used differently depending on whether the random variable is discrete or continuous:
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Discrete Random Variable: For a discrete random variable, the PDF is the probability that the variable takes on a specific value. The sum of all probabilities across the possible values equals one.
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Continuous Random Variable: For a continuous random variable, the PDF is represented by a curve, and the probability that the variable falls within a particular range is given by the area under the curve within that range. The total area under the curve for all possible values is equal to one.
Examples
Example 1: Discrete Random Variable
Suppose we roll a fair six-sided die. The discrete probability density function for this random variable, \(X\), where \(X\) is the outcome of the roll, is given by: \[ P(X=x) = \frac{1}{6} \quad \text{for } x \in {1, 2, 3, 4, 5, 6} \]
Example 2: Continuous Random Variable
Consider the continuous random variable \(X\) representing the heights of adult women in a certain population. The PDF might be described by a normal distribution: \[ f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} \] where \(\mu\) is the mean height, and \(\sigma\) is the standard deviation.
Frequently Asked Questions (FAQs)
Q1: What is the difference between a PMF and a PDF?
A: A PMF (probability mass function) is used for discrete random variables and gives the probability of the variable going through a specific value. A PDF is used for continuous random variables and describes the relative likelihood of the variable being close to a particular value.
Q2: How can you find the probability of a continuous random variable lying within a certain range?
A: For continuous random variables, you find the probability by calculating the area under the PDF curve over that range.
Q3: Why can’t we directly find the probability of a continuous random variable taking a specific value from its PDF?
A: For continuous random variables, the probability of taking an exact value is zero. Instead, probabilities are defined over intervals.
Q4: What properties must a PDF satisfy?
A: For both discrete and continuous PDFs:
- The function must be non-negative.
- The sum (for PMFs) or the area under the curve (for PDFs) must be equal to one.
Q5: Can a PDF have values greater than one?
A: Yes, but only for continuous random variables and within certain intervals, as long as the total area under the curve over all possible values is equal to one.
Related Terms
Cumulative Distribution Function (CDF)
A function describing the probability that a random variable \(X\) will take a value less than or equal to \(x\).
Expected Value
The long-run average value of repetitions of the experiment it represents, calculated as the sum of all possible values weighted by their probabilities.
Normal Distribution
A continuous probability distribution characterized by its bell-shaped curve, defined by its mean (µ) and variance (σ²).
Variance
A measure of the dispersion of a set of values, representing the average of the squared deviations from the mean.
Standard Deviation
The square root of the variance, representing the dispersion of a dataset.
Online References
- Khan Academy on PDFs
- Wolfram MathWorld: Probability Density Function
- Wikipedia: Probability Density Function
Suggested Books for Further Studies
- “Probability and Statistics for Engineers and Scientists” by Ronald E. Walpole, Raymond H. Myers
- “Introduction to Probability Models” by Sheldon Ross
- “Statistics for Business and Economics” by Paul Newbold, William L. Carlson, Betty Thorne
Fundamentals of Probability Density Function: Statistics Basics Quiz
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