GATE 2023 EC – Question 36
A random variable $X$, distributed normally as $N(0,1)$, undergoes the transformation $Y=h(X)$, given in the figure. The form of the probability density function of $Y$ is
(In the options given below, $a,b,c$ are non-zero constants and $g(y)$ is piece-wise continuous function)
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Correct answer: (B) $a\delta(y+1)+b\delta(y)+c\delta(y-1)+g(y)$
Explanation
$h(x)=0$ for $|x|\le1$, which has non-zero probability, so $Y=0$ with a finite probability: an impulse $b\delta(y)$. $h(x)=-1$ for $x\le-2$ and $+1$ for $x\ge2$, again with non-zero probability: impulses $a\delta(y+1)$ and $c\delta(y-1)$. For $1<|x|<2$ the ramp maps $X$ continuously onto $(-1,1)$, giving the continuous part $g(y)$.