ECE 461 Fall 2006August 24, 2006HOMEWORK ASSIGNMENT 1Reading: Text, Chapter 2Due Date: August 31, 2006 (in class)1. Use your knowledge of Gaussian and jointly Gaussian pdfs to get the answers to thefollowing directly (without resorting to integration).(a) Find the variance of the random variable that has densityfX(x) =1√4πe−(x−3)24, for all x.(b) Suppose fX,Y(x, y) =12πλ2e−x2+y22λ2. Find E[X2+ Y2].2. Let X1, X2, . . . , Xnbe i.i.d. random variables each with pdf fX(x).(a) Find the pdf of Y = min{X1, X2, . . . , Xn}.(b) Find the pdf of Z = max{X1, X2, . . . , Xn}.3. Random variables X and Y are jointly Gaussian with means mX= 1, mY= 2,variances σ2X= 4, σ2Y= 9, and Cov(X, Y ) = −4.(a) Find the correlation coefficient between X and Y .(b) If Z = 2X + Y and W = X − 2Y , find Cov(Z, W ).(c) Find the pdf of Z.(d) Find the joint pdf of Z and W .4. Suppose X is a random variable that is uniformly distributed on the interval [0, 1],that is fX(x) is 1 on the interval [0, 1] and 0 otherwise.(a) Suppose Y = e−2X. Find FY(y) and fY(y).Hint: Begin with FY(y) = P{Y ≤ y} = P{e−2X≤ y} = P{X ≥ −(log y)/2}.(b) Now suppose that a random process (which is only defined for t > 0) is g iven byY (t) = e−tX. Find the cdf and pdf of the random variable Y (t0), where t0is afixed positive number.cV.V. Veeravalli, 2006 15. Bounds on the Q function.Q(x) =Z∞xe−t2/2√2πdt(a) For x > 0 show that the following upper and lower bounds hold for the Q function:1 −1x2e−x2/2x√2π≤ Q(x) ≤e−x2/2x√2πHint: For the upper bound, write the integrand as a product of 1/t and te−t2/2,use integration by part s, and bound. For the lower bo und, integrate by parts oncemore and bound.(b) As you know from ECE 459 or a n equivalent communications course, the bit errorprobability for BPSK signaling in additive white Gaussian noise (AWGN) withPSD N0/2 is given by:Pe= Q r2EbN0!where Ebis the bit energy.Plot the error probabililty Pe(on a log scale) versus signal-to-noise ratio Eb/N0(in dB) using Matlab or Mathematica. (You may need to use an appropriatelymodified version of the error function in these packages.) Consider Eb/N0rangingfrom −5 dB to 15 dB. Also plot the bounds and compare.cV.V. Veeravalli, 2006