E1. Gaussian random variable W is known to have P[W>4] = 1/2. Based on this information, (a) sketch its possible PDF. (b) Now also assume that Var[W] is very small. Refine your sketch in the first part taking into account this new information. E2 We are told that r.v. X has mean value = 10 and variance = 2/3. Draw the pdf of X under the assumption that (a) X is Gaussian, (b) X is continuous and uniform. E3. What would you predict to be the minimum entropy (in bits) of a Bernoulli random variable? Explain. E4. An exponential r.v. B is known to have mean value 5. Write a formula for its PDF: E5. Sketch the approximate PDFs for continuous, zero-mean uniform random variable X when given that (a) E[X2] = 1, and (b) E[X2] = 2. E6. A noisy communication channel distorts a transmitted sequence of pulses (at amplitudes A) by adding to it samples of random variable N (having zero mean and standard deviation N). We learned that an estimated probability of bit transmission error in the channel is Q(A/N). What assumption about r.v. N were made in order to arrive at this estimate? E7. Give a strategy for estimating the CDF of random variable X, when the PDF of X is not known. E8. Samples of a zero-mean, uniformly-distributed continuous r.v. X are passed through a uniform quantizer to produce r.v. Y. (The quantizer range matches the range of X.) What will be the quantization signal-to-noise ratio, in dB, when it is given that there are 100 quantizer output levels? E9. Discrete r.v. Y = 2X 2, where r.v. X = Bernoulli(1/4). Find Cov[X,Y] by directly evaluating the expectation defining covariance, and then doublecheck your result using an alternate method. k1- You are tasked with estimating the acoustical impulse response of the human lungs (from the mouth to the back of the thorax). Describe a method to do this without exposing the human test subject to excessive levels of acoustical energy. K2. A Matlab function that generates random numbers, supposedly uniformly distributed over the range [0, 1], was written by your colleague and you are asked to verify that the underlying PDF is as specified. Formulate an experimental plan, including how you would verify that the experimental plan is correct, to test your colleague’s Matlab code. K3. Write Matlab code to generate 10^6 samples of a continuous r.v. X that is uniform over [0,5] Volts, and then measure and present the following data: estimated mean, estimated mean-square. K4. Given input array “User_Data”, a Matlab procedure returns the fraction of values in the array that lie below zero. How may you apply this procedure to estimate the CDF of the random variable whose samples are in “User_Data”? K5. Write code in Matlab that generates 10^6 samples of a Bernoulli(p) random variable. K6. As a student on a budget, based on our class discussions, (a) what is the minimum-cost solution to performing experiments on a PC using numerical computation software at UIC? (b) What features of your desktop PC would make such experimentation more capable? K7. We are told that the input signal to an amplitude quantizer is zero mean Gaussian, and that the output amplitude levels are all equally probable. What does this tell you about the distribution of these quantizer levels? K8. It is known that when estimating variance of a random variable X from N of its samples, the formula “Ave((X – Ave(X))2)” is biased to produce a lower result than the true value of Var[X], which is due to the error introduced by estimating Ave(X) {“Ave” operation is the average of the quantity that follows over N samples}. Describe how one may correct for this bias. K9. Explain two different methods to estimate the value of Q(z) for z>6 (outside of the range of the textbook table).

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