A possible probability density function (PDF) is any function, f(x), that satisfies two crucial criteria: it must be non-negative for all values of x, and the total area under the curve must equal 1. These conditions ensure that the PDF accurately represents the probabilities of a continuous random variable.
Defining a Probability Density Function (PDF)
A Probability Density Function (PDF) is a function that 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 value of the PDF at a specific point is not the probability of that value occurring. Instead, the probability is given by the integral of the PDF over a specific interval.
Conditions for a Valid PDF
To be a valid PDF, a function f(x) must satisfy the following two conditions:
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Non-negativity: f(x) ≥ 0 for all x ∈ ℝ (i.e., the function's value is greater than or equal to zero for all real numbers). This ensures that the 'likelihood' of observing any value is never negative.
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Normalization: ∫−∞∞ f(x) dx = 1 (i.e., the integral of the function over its entire domain is equal to 1). This ensures that the total probability of observing some value of the random variable is equal to 1.
Examples of Probability Density Functions
Here are a few common examples of probability density functions:
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Uniform Distribution:
- f(x) = 1/(b-a) for a ≤ x ≤ b and f(x) = 0 otherwise. Here, a and b are constants.
- This represents a situation where all values between a and b are equally likely.
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Normal Distribution (Gaussian Distribution):
- f(x) = (1 / (σ√(2π))) e^(-((x - μ)² / (2σ²)))*.
- Where μ is the mean and σ is the standard deviation. This is a bell-shaped curve, ubiquitous in statistics.
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Exponential Distribution:
- f(x) = λe^(-λx) for x ≥ 0 and f(x) = 0 otherwise. Where λ > 0 is the rate parameter.
- This often models the time until an event occurs.
Why these conditions matter
- Non-negativity: If a function had negative values, it would imply negative probabilities, which are not meaningful in probability theory.
- Normalization: The total probability of any outcome happening must be 1 (or 100%). The integral represents the sum of all possible probabilities.
In Summary
A valid probability density function (PDF) must be non-negative and integrate to 1 over its entire domain. These conditions ensure it can be interpreted as probabilities associated with a continuous random variable. Many different functions can serve as PDFs, depending on the specific distribution they represent.
