9.5 Probability Density Function (PDF) Formula

The Probability Density Function (PDF) is a fundamental concept in probability theory and statistics, particularly for continuous random variables. It describes the relative likelihood for a continuous random variable to take on a given value.

The Formula

For a continuous random variable $X$, its probability density function, denoted as $f(x)$, is defined such that the probability of $X$ falling within a specific interval $[a, b]$ is given by the integral of $f(x)$ over that interval:

$P(a \leq X \leq b) = \int_{a}^{b} f(x) , dx$

Explanation of the Formula

• $P(a \leq X \leq b)$: This represents the probability that the continuous random variable $X$ will take on a value between $a$ and $b$, inclusive.
• $\int_{a}^{b}$: This is the mathematical symbol for a definite integral. It signifies summing up the infinitesimally small contributions of $f(x)$ across the interval from $a$ to $b$.
• $f(x)$: This is the Probability Density Function (PDF) of the random variable $X$. It's a function of $x$, the possible values the random variable can take.
• $dx$: This indicates that the integration is performed with respect to the variable $x$.

Key Properties of a PDF

For $f(x)$ to be a valid PDF, it must satisfy two main conditions:

1. Non-negativity: The PDF must be non-negative for all possible values of $x$: $f(x) \geq 0$ for all $x$

2. Total Probability: The total area under the curve of the PDF over its entire domain must equal 1. This signifies that the probability of the random variable taking on any value within its possible range is 1 (or 100%). $\int_{-\infty}^{\infty} f(x) , dx = 1$

Interpretation of PDF Values

It's important to understand that the value of $f(x)$ at a specific point $x$ is not a probability. Instead, it represents the density of probability around that point. The higher the value of $f(x)$, the more likely it is for the random variable to fall within a small interval around $x$.

Example: Exponential Distribution

Consider an exponential distribution, often used to model the time until an event occurs. Its PDF is:

$f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$, and $f(x) = 0$ for $x < 0$.

Here, $\lambda$ is the rate parameter.

To find the probability that the event occurs between time $a$ and time $b$ (where $a < b$ and both are non-negative):

$P(a \leq X \leq b) = \int_{a}^{b} \lambda e^{-\lambda x} , dx$

Solving this integral:

$P(a \leq X \leq b) = [-e^{-\lambda x}]_{a}^{b}$ $P(a \leq X \leq b) = (-e^{-\lambda b}) - (-e^{-\lambda a})$ $P(a \leq X \leq b) = e^{-\lambda a} - e^{-\lambda b}$

This formula allows us to calculate the probability of the random variable falling within any given interval using its PDF.