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I am reading a text that states that with a discrete random variable the probability mass function is used since x can take on particular values with a particular probability. However, it goes on to say, with a continuous case since the probability of a particular value of X is not measurable instead a PDF is used where a “probability mass” per unit per length around x can be measured. I think it’s the use of the word “mass” to describe a PDF is throwing me off. What are your thoughts?

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A PDF $f$ for a continuous random variable $X$ satisfies $P(a \le X \le a+\delta) = \int_a^{a+\delta} f(x) \, dx$ for $\delta > 0$.

If $\delta$ is small, then the integral is $\approx \delta f(a)$ (the area under the graph of $f$ on the interval $[a, a+\delta]$ is approximately the area of the rectangle with width $\delta$ and height $f(a)$). Rearranging yields $$f(a) \approx \frac{P(a \le X \le a+\delta)}{\delta}.$$

The fraction on the right-hand side is the "probability mass per unit length around $a$." I would think of "probability mass" of an event simply as the probability of the event.

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