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