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I'd like a steer in the right direction with this problem.

I would like to show that

[tex]P\{x_1\leq X \leq x_2\}=F_{X}(x_2)-F_{X}(x_1^{-})\quad(1)[/tex]

Where:

[tex]X[/tex] is a random variable.

[tex]F_{X}(x) \equiv P\{X \leq x \} [/tex] is its cumulative distribution function.

My notes only give an example (using dice) to show that this is true.

Generally

[tex]P\{x_1 < X \leq x_2\}=F_{X}(x_2)-F_{X}(x_1)\quad(2)[/tex]

and

[tex]P\{X = x_2\}=F_{X}(x_2)-F_{X}(x_2^{-})\quad (3)[/tex]

the latter of which is easy to prove.

I've been trying to rewrite (1) in terms of (2) & (3) but have had no success so far.

Any ideas would be most welcomed

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# Proof with regards to cumulative distribution function

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