Why does the expectation of a continuous random variable use integration instead of summation?

Conditions

• X is a continuous random variable
• $f(x)$ is the probability density function
• The integral converges absolutely

Reasoning, step by step

1. Contrast discrete variables (countable values) with continuous variables (uncountable values).
2. Recognize that summation $\sum x_k p_k$ works for discrete cases.
3. Understand that for continuous cases, probability is distributed over intervals, described by density $f(x)$.
4. Replace the discrete probability $p_k$ with the infinitesimal probability $f(x)dx$.
5. Replace the summation symbol with the integral symbol to accumulate over the continuum.

Example

The video shows the transition from $E(X)=\sum_{k=1}^\infty x_k p_k$ for discrete variables to $E(X)=\int_{-\infty}^{\infty} x f(x) dx$ for continuous variables, explaining the latter as an improper integral.

Common misconceptions

• Believing that continuous variables have a countable list of values with non-zero probabilities.
• Thinking that integration is just a different notation for summation without conceptual difference in the nature of the variable.
• Confusing probability density $f(x)$ with probability $P(X=x)$.

Watch the explanation

Connected concepts

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• What is the role of $f(x)dx$ in the expectation integral?
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• Why is absolute convergence required for the continuous expectation?