ECE Ph.D. Qualifying Exam

Communication Signal (CS)

Question 1: Random Variable

August 2016 Problem 1


Solution

a)
$ F_x(\omega)= \begin{cases} 0 & \omega<0 \\ \dfrac{1}{2}hb-\dfrac{1}{2}hb(\dfrac{h-\omega}{h})^2=\dfrac{2\omega}{h}-\dfrac{w^2}{h^2} & 0\le\omega<h \\ 1 & \omega\ge h \end{cases} $

b)
$ f_x(\omega)=\dfrac{\partial F_x(\omega)}{\partial\omega} $
$ f_x(\omega)= \begin{cases} 0 & \omega<0 \\ \dfrac{-2}{h^2}\omega+\dfrac{2}{h} & 0\le\omega<h \\ 0 & \omega\ge h \end{cases} $

c)
$ X(\omega)\bar=\int_{-\infty}^{\infty} \omega f_x(\omega) dx =\int_{0}^{h} -\dfrac{2}{h^2}\omega^2 +\dfrac{2}{h}\omega d\omega =\dfrac{1}{3}h $

d)
$ P(x>\dfrac{h}{3})=\int_{\dfrac{h}{3}}^{+\infty} \omega f_x(\omega)dx = \int_{\dfrac{h}{3}}^{h} -\dfrac{2}{h^2}\omega+\dfrac{2}{h}d\omega =\dfrac{4}{9} $


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