Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2006

1 (33 points)

Let $\mathbf{X}$ and $\mathbf{Y}$ be two joinly distributed random variables having joint pdf

$f_{\mathbf{XY}}\left(x,y\right)=\left\{ \begin{array}{lll} 1, & & \text{ for }0\leq x\leq1\text{ and }0\leq y\leq1\\ 0, & & \text{ elsewhere. } \end{array}\right.$

(a)

Are $\mathbf{X}$ and $\mathbf{Y}$ statistically independent? Justify your answer.

$f_{\mathbf{X}}\left(x\right)=\int_{-\infty}^{\infty}f_{\mathbf{XY}}\left(x,y\right)dy=\int_{0}^{1}dy=1\text{ for }0\leq x\leq1.$

$f_{\mathbf{Y}}\left(y\right)=\int_{-\infty}^{\infty}f_{\mathbf{XY}}\left(x,y\right)dx=\int_{0}^{1}dx=1\text{ for }0\leq y\leq1.$

Since $f_{\mathbf{XY}}\left(x,y\right)=f_{\mathbf{X}}\left(x\right)f_{\mathbf{Y}}\left(y\right)$ , $\mathbf{X}$ and $\mathbf{Y}$ are $statistically independent.$

(b)

Let $\mathbf{Z}$ be a new random variable defined as $\mathbf{Z}=\mathbf{X}+\mathbf{Y}$ . Find the cdf of $\mathbf{Z}$ .

$F_{\mathbf{Z}}\left(z\right)=P\left(\left\{ \mathbf{Z}\leq z\right\} \right)=P\left(\left\{ \mathbf{X}+\mathbf{Y}\leq z\right\} \right).$

• i) if $z<0$ , then $F_{\mathbf{Z}}\left(z\right)=0$ .

• ii) if $z\geq2$ , then $F_{\mathbf{Z}}\left(z\right)=1$ .

• iii) if $0\leq z\leq1$ , then $F_{\mathbf{Z}}\left(z\right)=\iint f_{\mathbf{XY}}\left(x,y\right)dxdy=\iint1\cdot dxdy=\frac{1}{2}z^{2}$ .

• iv) if $1<z<2$ , then $F_{\mathbf{Z}}=\iint f_{\mathbf{XY}}\left(x,y\right)dxdy=\iint1\cdot dxdy=1-\frac{1}{2}\left(2-z\right)^{2}$ .

$\therefore F_{\mathbf{Z}}\left(z\right)=\left\{ \begin{array}{lll} 0 & & ,z<0\\ \frac{1}{2}z^{2} & & ,0\leq z\leq1\\ 1-\frac{1}{2}\left(2-z\right)^{2} & & ,1<z<2\\ 1 & & ,z\geq2 \end{array}\right.$

(c)

Find the variance of $\mathbf{Z}$ .

$f_{\mathbf{Z}}\left(z\right)=\left\{ \begin{array}{lll} z & & ,0\leq z\leq1\\ 2-z & & ,1<z<2\\ 0 & & \text{,otherwise.} \end{array}\right.$

$E\left[\mathbf{Z}\right]=\int_{-\infty}^{\infty}z\cdot f_{\mathbf{Z}}\left(z\right)dz=\int_{0}^{1}z^{2}dz+\int_{1}^{2}\left(2z-z^{2}\right)dz=\frac{1}{3}z^{3}\Bigl|_{0}^{1}+z^{2}-\frac{1}{3}z^{3}\Bigl|_{1}^{2}=\frac{1}{3}+3-\frac{7}{3}=1.$ $E\left[\mathbf{Z}^{2}\right]=\int_{-\infty}^{\infty}z^{2}\cdot f_{\mathbf{Z}}\left(z\right)dz=\int_{0}^{1}z^{3}dz+\int_{1}^{2}\left(2z^{2}-z^{3}\right)dz=\frac{1}{4}z^{4}\Bigl|_{0}^{1}+\frac{2}{3}z^{3}-\frac{1}{4}z^{4}\Bigl|_{1}^{2}=\frac{1}{4}+\frac{14}{3}-\frac{15}{4}=\frac{7}{6}.$ $Var\left[\mathbf{Z}\right]=E\left[\mathbf{Z}^{2}\right]-\left(E\left[\mathbf{Z}\right]\right)^{2}=\frac{1}{6}.$

## Alumni Liaison

Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal