The integral of the probability density function (PDF) over its entire range is always equal to 1.
Understanding Probability Density Functions (PDFs)
A probability density function, often denoted as f(x), describes the relative likelihood for this random variable to take on a given value. Key characteristics of a PDF include:
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Non-negativity: The value of the PDF is always greater than or equal to zero for all possible values of the random variable. This reflects the fact that probability cannot be negative.
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Area under the curve: The total area under the curve of the PDF, calculated by integrating the function over its entire range, must equal 1. This represents the certainty that the random variable will take on some value within its possible range. Mathematically, this is expressed as:
∫∞-∞ f(x) dx = 1
Why the Integral Equals 1
The integral of the PDF represents the cumulative probability over the entire range of the random variable. Since the random variable must take on some value within its defined range, the probability of this occurring is 1, or 100%. Therefore, the area under the entire PDF curve must equal 1.
Example
Imagine a PDF that describes the distribution of heights in a population. Integrating the PDF from the smallest possible height to the largest possible height will give you the probability that someone's height falls within that range. Since everyone must have a height within that range, the result of the integral is 1.
Implications
The fact that the integral of a PDF equals 1 is fundamental to probability theory and statistics. It ensures that the probabilities calculated using the PDF are consistent and meaningful. If the integral were not equal to 1, the function would not be a valid probability density function.
