A probability density function (PDF) must satisfy two fundamental properties to be valid.
A probability density function (PDF), often denoted as f(x), describes the likelihood of a continuous random variable taking on a given value. Unlike probability mass functions for discrete variables, the value of the PDF at a specific point x does not directly represent a probability. Instead, the probability is found by integrating the PDF over a range of values.
To accurately represent probabilities for a continuous distribution, a function must adhere to the following conditions:
Essential Properties of a Probability Density Function
Based on the provided reference, a probability density function f(x) must satisfy the following conditions:
- Non-Negativity: The function's value must be greater than or equal to zero for all possible outcomes.
- Condition: f(x) ≥ 0 for all x ∈ R
- Explanation: Probabilities cannot be negative. While f(x) isn't a direct probability, its non-negativity ensures that when you integrate over an interval to find a probability, the result will also be non-negative.
- Total Probability Must Equal One: The total area under the curve of the probability density function over its entire domain must be exactly equal to 1.
- Condition: ∞∫−∞f(x)dx=1
- Explanation: This condition signifies that the probability of the continuous random variable taking any value within its entire range is 100%, or 1. This is analogous to the sum of all probabilities in a discrete distribution equaling 1.
Why These Properties Are Crucial
These two properties are foundational because they ensure that the function f(x) behaves like a valid probability measure when integrated over any interval [a, b]. The probability P(a ≤ X ≤ b) is calculated as the integral ∫ab f(x) dx.
- The non-negativity (f(x) ≥ 0) ensures that the calculated probability P(a ≤ X ≤ b) is always non-negative.
- The total integral of 1 ensures that the probability of the variable falling within any possible range is consistently scaled so that the total probability space is 1.
Summary Table
| Property | Condition | Description |
|---|---|---|
| Non-Negativity | f(x) ≥ 0 for all x | The function's value is never negative. |
| Total Area Under Curve | ∫−∞^∞ f(x) dx = 1 | The integral over the entire domain equals one. |
Adherence to these two rules is essential for any function to be considered a valid probability density function describing a continuous probability distribution.
