Question

1. Find Fx in terms of φ (t). Is X a continuous random variable ? 2. Compute p(X 0) 3. Compute E(X). Hint: use the CDF expectation formula, and integration by parts. You may assume that lim, t"o(-t) 0 for all n 2 0. 4. Find the CDF Fx (u) 5. Compute V(X). Hint: use Fxa, and follow the same hint of part (3)

Answer 1

For $Z\sim N\left ( 0,1 \right )$ and $X=\max\left ( Z,0 \right )$ . We have $X=\left\{\begin{matrix} 0; & Z<0 \\ Z; & Z\geqslant 0 \end{matrix}\right.$

a) The CDF is

$F_X\left ( 0 \right )=P\left ( Z<0 \right )=\Phi \left ( 0 \right )=0.5\\ F_X\left ( x \right )=P\left ( X=0 \right )+P\left ( 0<Z<z \right )\\ F_X\left ( x \right )=P\left ( Z<0 \right )+P\left ( 0\leqslant Z<z \right )=\Phi \left ( z \right )\\ F_X\left ( x \right )=\Phi \left ( x \right );x>0$

To sum up,

${\color{Blue} F_X\left ( x \right )=\left\{\begin{matrix} 0.5; & X=0\\ \Phi \left ( x \right ) ;& x>0 \end{matrix}\right.}$

X is a mixed (discrete + continuous random variable)

b) From part (a), ${\color{Blue} p\left ( X=0 \right )=0.5}$ .

c) The expected value is

$E\left ( X \right )=0\times P\left ( X=0 \right )+\int_{0}^{\infty }\frac{x}{\sqrt{2\pi }}e^{-x^2/2}dx\\ E\left ( X \right )=0+\frac{1}{\sqrt{2\pi }}\left [-e^{-x^2/2} \right ]_{0}^{\infty }\\ {\color{Blue} E\left ( X \right )=\frac{1}{\sqrt{2\pi }}}$

d) The CDF

$F_{X^2}\left ( u \right )=P\left ( X^2<u \right )\\ F_{X^2}\left ( u \right )=P\left ( X<\sqrt{u} \right )\\ {\color{Blue} F_{X^2}\left ( u \right )=\Phi \left ( \sqrt{u} \right );u>0}$

e)

$E\left ( X ^2\right )=0^2\times P\left ( X=0 \right )+\int_{0}^{\infty }\frac{x^2}{\sqrt{2\pi }}e^{-x^2/2}dx\\ E\left ( X^2 \right )=\sqrt{\frac{\pi }{2}}$

The variance is

$Var\left ( X \right )=E\left ( X ^2\right )-E\left ( X \right )^2\\ Var\left ( X \right )=E\left ( X ^2\right )-E\left ( X \right )^2\\ Var\left ( X \right )=\sqrt{\frac{\pi }{2}}-\frac{1}{2\pi }$