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Hi lets say your CDF is from $Unif(-1,1)$ so $F(x) = (x+1)/2$

Its easy to understand how $$P(X<x) = F(x) \implies xP(X>x) = 1 - F(x)$$.

But how do I breakdown $P(|X| < x)$ or $P(X^2 < x)$? What about $P(|X| > x)$ or $P(X^2 > x)$?

Sycorax
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hxlaclhemy
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  • Hint: For $x > 0$, $P(|X| < x) = P(-x < X < x)$. Do you know how to determine the value of $P(-x < X < x)$ from the CDF? Also, please use MathJax to format your equations. – Dilip Sarwate Nov 18 '22 at 22:37
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    I strongly suggest drawing a few pictures, to satisfy yourself how an interval or set of intervals/half intervals for $X$ on the real line will correspond to some condition like $|X|<x$. Draw, draw, draw until the ideas are clear enough that you no longer need the picture to understand what's happening. – Glen_b Nov 19 '22 at 01:11
  • @glen_B, I think that is the best approach! – hxlaclhemy Nov 23 '22 at 02:19
  • See https://stats.stackexchange.com/questions/138763/pdf-of-function-of-x/138922#138922 for a worked example. – whuber Nov 23 '22 at 15:14

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