Consider the random variables (RVs) X1, X2, X3 and X4, that are independent and identically distributed (IID) with uniform distribution in the interval (-2, 2). (a) Demonstrate that the probability density function (PDF) fz (2) of Z = X1 + X2 can be obtained as the convolution between the PDFs of X1 and X2. (b) Use MATLAB and the convolution function conv . m to find and plot the PDF fz (z) of Z Hint: To compute the continous time (CT) y(t) = x(t)*h(t) in discrete time (DT) - Obtain x[n] = x(nTs) and h[n]=h(nT's) from the CT signals x(t) and h(t) using sampling period Is. - Obtain the DT output signal y [n]= x[n]*h[n]using conv .m - Obtain the CT reconstructed signal yr(nT's) = Ts y[n]; then y(nT's) =yr(nT's) can be exactly recovered at its samples. Note: To avoid aliasing (loss of information), the sampling frequency f, =1/T, is selected to be at least twice that of the maximum frequency of y(t). (c) Repeat (b) to find and plot the PDF fR(r) of R= X1 + X2 + X3 =Z + X3. (d) Repeat (b) to find and plot the PDF fo(q) of Q= X1 + X2 + X3 + X4 = R + X4. (e) Using the central limit theorem, find the mean and variance of Q = X1 + X2 + X3 + X4; use these values and the MATLAB command normpdf . m to find and plot an approximated PDF of Q. On the same axis (superimposed), provide the plot of this PDF and the one obtained in (d). Discuss yours observations