# random variable in an interval

I have a random variable $X$. The constants $a$, $b$ and $c$ are given. I have to find the interval $I$ such that $P(a\in (X-b,X+b))=c$. My question is actually not how to calculate this interval. How should I think of a random variable in an interval? Is there an intuitive way?

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how does interval $I$ play a role here? –  Ilya Jan 21 '13 at 14:40
@Ilya I didnt mention it well but I meant $I=(X-b,X+b)$. –  Badshah Jan 21 '13 at 14:46

## 2 Answers

$$[a\in(X-b,X+b)]=[a-b\lt X\lt a+b]=[X\in(a-b,a+b)]$$

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okey thanks, and how should this help me find the interval $I=(X-b,X+b)$? because I dont see why having $P(X\in (a-b,a+b))=c$ should give me $I$. –  Badshah Jan 21 '13 at 14:57

$a\in(X-b,X+b)\implies X\in(a-b,a+b)$

and $P((a-b)<X<(a+b))=F(a+b)-F(a-b)$ where F is the cumulative distribution function of random variable $X$

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