How do you derive CDF from a PDF?

Q: How do you derive CDF from a PDF?

This question evaluates understanding of the relationship between a probability density function and its cumulative distribution function, the formal properties that PDFs and CDFs must satisfy, and competence with piecewise definitions and numerical approximation considerations.

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You are given a continuous random variable $X$ with probability density function (PDF) $f_X(x)$ .

Write the definition of the cumulative distribution function (CDF) $F_X(x)$ in terms of $f_X(x)$ .
State the conditions a valid PDF and CDF must satisfy.
If $f_X(x)$ is only defined piecewise (different formulas on different intervals), explain how you would compute $F_X(x)$ correctly across all ranges of $x$ .
(Practical) If $f_X(x)$ has no closed-form integral, describe a reasonable numerical approach to approximate $F_X(x)$ for a given $x$ , and how you’d control error.

How do you derive CDF from a PDF?

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