Question

7. Let X be a random variable with distribution function Fx. Let a < b. Consider the following 'truncated' random variable Y: if X < a, if X > b. (a) Find the distribution function of Y in terms of Fx. (It will be a good additional exercise to sketch FY though you don't have to hand it in.) (b) Evaluate the limit lim FY (y) b-00

Answer 1

Now the PDF of the random variable $Y=g\left ( X \right )$ is

$f_Y\left ( y \right )=f_X\left ( g^{-1}\left ( y \right ) \right )\left | \frac{\mathrm{d} g^{-1}\left ( y \right ) }{\mathrm{d} y} \right |$

a)For the given random variable $Y=g\left ( X \right )$ ,

$f_Y\left ( y \right )=\left\{\begin{matrix} F_X\left ( a \right ); & y<a \\ f_X\left ( g^{-1}\left ( y \right ) \right )\left | \frac{\mathrm{d} g^{-1}\left ( y \right ) }{\mathrm{d} y} \right |; &a\leqslant x\leqslant b \\ 1; & y>b \end{matrix}\right.$

The distribution function of $Y=g\left ( X \right )$ is $F_Y\left ( y \right )=\int_{-\infty }^{y} f_Y\left ( y \right )dy$

${\color{Blue} f_Y\left ( y \right )=\left\{\begin{matrix} F_X\left ( a \right ) & y<a \\F_X\left ( b \right )-F_X\left ( y \right ); &a\leqslant y\leqslant b \\ F_X\left ( b \right ); & y>b \end{matrix}\right.}$

b)The upper and lower limits of any distribution function is $0$ and $1$.

$\lim_{a\rightarrow -\infty }F_Y\left ( a \right )=F_Y\left ( -\infty \right )=0\\ \lim_{b\rightarrow \infty }F_Y\left ( b \right )=F_Y\left ( \infty \right )=1$