How does the video approximate the probability of a continuous random variable falling into a small interval?

Conditions

• The interval $(x, x+\Delta x]$ is very small
• Higher-order infinitesimals are neglected
• $f(x)$ is the probability density function

Reasoning, step by step

1. Identify a small interval of width $\Delta x$ on the real line.
2. Select a representative point $x$ (or $\xi_i$) within this interval.
3. Evaluate the probability density $f(x)$ at this point.
4. Multiply the density value by the interval width $\Delta x$.
5. Interpret the product $f(x)\Delta x$ as the approximate probability of $X$ falling within that interval.

Example

The video displays the formula: "From equation (4.2), we know that if higher-order infinitesimals are neglected, $P{x < X \le x + Δx}$ ≈ $f(x)Δx$."

Common misconceptions

• Believing that $f(x)$ itself is the probability at point $x$.
• Thinking that the probability of a continuous random variable taking an exact single value is non-zero.
• Confusing probability density with cumulative probability.

Watch the explanation

Connected concepts

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• What is the role of $f(\xi_i)\Delta x_i$ in the expectation integral?
• Why is the probability of a continuous random variable at a single point zero?
• How does this approximation lead to the definition of expectation as an integral?