# How distribution function behaves when $b\to\infty$

Let $X$ be a r.v. with dist func $F$ and let $a < b$ .Find and sketch the distribution function of

$$z=\begin{cases} x & \text{if } |x| \leq b \\ 0 & \text{if } |x| > b. \end{cases}$$

How does this distribution function behave as $b \rightarrow \infty$?

I do not understand what 'a' have to do with it? Does the graph meets the x-axis at a and is it like z=x line...

Thanks

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Given the problem as stated, $a$ has nothing to do with it. Is there another part to the problem? –  Jonathan Christensen Oct 29 '12 at 19:47
So How should I sketch the graph ? and limit will go to 1 , right ? –  idobi182 Oct 29 '12 at 19:52
No, the limit of the random variables $Z_b$ will be $X$ itself, and their distribution function tend to $F$. –  Berci Oct 29 '12 at 20:02
I see..But what about the graph? Will it pass from the origin? Will it be like z=x after x>b and x<-b will it be 0 ? –  idobi182 Oct 29 '12 at 20:13
If you are talking about the graph of $z$ in terms of $x$, it will be flat (equal to zero) between $-b$ and $b$, and $z=x$ outside of that interval. But the question doesn't ask anything about this graph, so I don't see how it's relevant. –  Jonathan Christensen Oct 30 '12 at 17:31