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I am trying to figure out where the cdf has a jump. I believe there is a jump at $x=\frac{1}{2}$, but it is obviously wrong and I am not sure why that is the case. I would appreciate if someone could provide an explanation on how it is done for any random variable and how the sketch looks like.

$Y(\omega)=\omega$ for $\omega\in[0,\frac{1}{2}]$ and $Y(\omega)=\frac{1}{4}$ for $\omega\in(\frac{1}{2},1]$

  • Hi! I am a bit confused about your notation. $Y$ is usually used for random variables, with $\omega$ being an element of the sample space. If so, we do not know anything about the CDF of $Y$, as its distribution is not given. Finally, the function $Y$ which you provide indeed has a jump at $1/2$, which you can see by plotting it. – Idontgetit Mar 11 '21 at 19:05
  • @Idontgetit Presumably the sample space here is $[0,1]$ with Lebesgue measure. The CDF has a jump at a value the random variable takes with nonzero probability. That value is $1/4$ in this case. – Robert Israel Mar 11 '21 at 19:11
  • @RobertIsrael, could you sketch this one? –  Mar 11 '21 at 19:17

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For $0 \le x \le 1/2$, $\mathbb P(\omega \le 1/2, X \le x) = \mathbb P(\omega \le x) = x$. Then there is an additional probability of $1/2$ that $X = 1/4$ with $\omega > 1/2$. So the cdf is $$ F(x) = \mathbb P(Y \le x) = \cases{0 & if $x < 0$\cr x & if $0 \le x < 1/4$\cr 1/2 + x & if $1/4 \le x \le 1/2$\cr 1 & if $x > 1/2$\cr}$$

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Robert Israel
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