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If $X$ and $Y$ are two independent random variables, with the following probability density functions:

$f_x(x)=1/2*e^{-|x|}$, $x \in R$ and $f_y(y)=e^{-y}$, $y>0$.

Find the cumulative distribution function of $Z=min\{X^2 - 1, 2Y+1\}$.

Any help with this would be really appreciated, since I do not understand what I am supposed to do here at all. Thanks

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  • $\begingroup$ Do you know the definition of cumulative distribution function? $\endgroup$
    – NCh
    May 10, 2017 at 18:16
  • $\begingroup$ @NCh yes, it's the integral of probability density function, from minus infinity to infinity $\endgroup$
    – ivana14
    May 10, 2017 at 18:18
  • $\begingroup$ You are not given probability density function of $Z$. See hint below for the definition. $\endgroup$
    – NCh
    May 10, 2017 at 18:38

1 Answer 1

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Hint: Notice that $\min\{a, b\} > c \iff a>c \text{ and } b>c$, and then consider $F_Z(z) = P(Z\leq z) = 1 -P(Z>z).$

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