Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2000

# Part 4

A RV is given by $\mathbf{Z}=\sum_{n=0}^{8}\mathbf{X}_{n}$ where $\mathbf{X}_{n}$ 's are i.i.d. RVs with characteristic function given by $\Phi_{\mathbf{X}}\left(\omega\right)=\frac{1}{1-j\omega/2}.$

(a) Determine the characteristic function of $\mathbf{Z}$ .

(b) Determine the pdf of $\mathbf{Z}$ . You can leave your answer in integral form.

# Solution 1 (retrived from here)

(a)

$\Phi_{\mathbf{Z}}\left(\omega\right)=E\left[e^{i\omega\mathbf{Z}}\right]=E\left[e^{i\omega\sum_{n=0}^{8}\mathbf{X}_{n}}\right]=E\left[\prod_{n=0}^{8}e^{i\omega\mathbf{X}_{n}}\right]=\prod_{n=0}^{8}E\left[e^{i\omega\mathbf{X}_{n}}\right]=\left(\frac{1}{1-j\omega/2}\right)^{9}.$

(b)

$f_{\mathbf{Z}}\left(z\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\Phi_{\mathbf{Z}}\left(\omega\right)e^{-i\omega z}d\omega=\frac{1}{2\pi}\int_{-\infty}^{\infty}\left(\frac{1}{1-j\omega/2}\right)^{9}e^{-i\omega z}d\omega.$

Write it here.

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