What is the relationship between the probability density function and the probability of a small interval?

Conditions

• The interval width $\Delta x$ is very small
• Higher-order infinitesimals are neglected
• $f(x)$ is continuous at the reference point

Reasoning, step by step

1. Define the probability density function $f(x)$.
2. Consider a small interval $(x, x+\Delta x]$.
3. Approximate the probability over this interval using the rectangle area under the density curve.
4. State the approximation formula: Probability $\approx$ Density $\times$ Width.
5. Explain that this approximation becomes exact in the limit as $\Delta x \to 0$ (leading to the integral).

Example

The video inserts a textbook page explaining: "This means that the probability of X falling in the small interval (x, $x+Δx$] is approximately equal to $f(x)Δx$."

Common misconceptions

• Believing that $f(x)$ is the probability at point $x$.
• Thinking that the probability of any interval is exactly $f(x)\Delta x$ regardless of interval size.
• Confusing the area under the density curve with the density value itself.

Watch the explanation

Connected concepts

Explore next

• How does this approximation help in understanding the expectation integral?
• Why is the probability at a single point zero for a continuous variable?
• What is the geometric interpretation of $f(x)\Delta x$?