Calculating Probabilities With Continuous Random Variables

For a continuous random variable with a known PDF, probabilities are computed as integrals of the PDF. This lesson covers interval probabilities, the fact that single points have probability zero, one-sided (half-line) probabilities via improper integrals, the complement rule, and probabilities under piecewise-defined PDFs.

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Tutorial

Probability as the Area Under the PDF

For a continuous random variable $X$ with probability density function $f(x),$ the probability that $X$ takes a value in the interval $[a, b]$ equals the area under $f$ over that interval:

P(a \le X \le b) = \int_a^b f(x)\, dx.

For example, suppose $X$ has PDF $f(x) = 2x$ for $0 \le x \le 1$ and $0$ elsewhere. Then

P\!\left(\tfrac{1}{4} \le X \le \tfrac{1}{2}\right) = \int_{1/4}^{1/2} 2x\, dx = \big[x^2\big]_{1/4}^{1/2} = \dfrac{1}{4} - \dfrac{1}{16} = \dfrac{3}{16}.