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The question is

$F$ is a distribution function. Show that $P\{X = x\} = F(x) - \lim_{y \uparrow x}F(y)$

This is trivial. But how can we prove it in a formal way?

1 Answers 1

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$F(y) = \Pr (X \le y)$ so $\lim_{y \uparrow x}F(y) =\lim_{y \uparrow x} \Pr (X \le y) = \Pr (X \lt x) $ and thus $F(x) - \lim_{y \uparrow x}F(y)= \Pr (X \le x)- \Pr (X \lt x)= \Pr (X = x).$

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    @Henry : When you write 5\lim_{y\to x} g(y) in a "displayed" (as opposed to "inline") setting, then it looks like this: $\displaystyle5\lim_{y\to x} g(y)$ with "$y\to x$" directly below "$\lim$", proper spacing before and after "$\lim$", and "$\lim$" not italicized. That is standard. But when you write 5 lim_{y\to x} g(y), with no backslash in \lim, then it looks like this: $\displaystyle 5 lim_{y\to x} g(y)$. (In an "inline" setting, the subscript $y\to x$ is not directly below "$\lim$" unless you use \displaystyle.)2012-05-06