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What is the difference between these two terms? From what I saw the two terms are not exactly interchangeable. I think that probability distribution implies probability measure, but the converse is not true. If this last statement is true, can you make an example?

Lex
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Suppose that $(\Omega, \mathcal{S}, P)$ is a probability space and that $X\rightarrow \mathbb{R}$ is a random variable. You can define a borel measure $P_X$ on $\mathbb{R}$ by

$$P_X(E) = P(X^{-1}(E)), \qquad E\in\mathcal{B}(\mathbb{R}).$$

This is the distribution of $X$. The cumulative distribution function is defined by

$$F_X(x) = P_X((-\infty, x]),\qquad x\in \mathbb{R}.$$

Sometimes, this is just called the distribution function. Every right continuous nondecreasing function $F:\mathbb{R}\rightarrow [0,1]$ with $F(\infty) = 1$ and $F(-\infty) = 0$ is a probability distribution. (for proof see Laha and Rohatgi's book).

ncmathsadist
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