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Suppose $X_1$ is a standard normal random variable. Define $X_2$ as $X_2$ $=$ $-X_1$ if $|X_1|<1$ , and $X_2$$=$$X_1$ , otherwise.

(a) Show that $X_2$ is also a standard normal random variable. (b) Obtain the cumulative distribution function of $X_1 + X_2$ in terms of the cumulative distribution function of a standard normal random variable.

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  • $\begingroup$ So, you are saying that $X_2=X_1$, because in $X_2=X_1$ in both conditions. Are you sure your wording is correct? $\endgroup$ – BlackMath Aug 25 '18 at 16:04
  • $\begingroup$ Sorry sorry....edited that typo $\endgroup$ – Prof.Shanku Aug 25 '18 at 16:06
  • $\begingroup$ Anything you tried? Same second question:math.stackexchange.com/questions/361212/… $\endgroup$ – StubbornAtom Aug 25 '18 at 16:37
  • $\begingroup$ Thanks! @stubbornAtom $\endgroup$ – Prof.Shanku Aug 25 '18 at 16:43
  • $\begingroup$ But how to prove part (a)? $\endgroup$ – Prof.Shanku Aug 25 '18 at 16:49

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