Question

Open book, open notes. No collaboration. Return this sheet along with your answers (17) 1. Assume that a binary communication system sends message "O" as -5 V and message l" as +5 V randomly but with a "I" three times as likely as a "O". Because of uniformly-distributed noise picked up during transmission, a "o" arrives at the receiver input as a voltage uniformly distributed between -7 V and -3 V, and a "" arrives at the receiver input as a voltage uniformly distributed between +3 V and +7 V. Assurme that a random experiment consists of measuring the voltage at the receiver input for a message transmission. Let this measured voltage be outcome x of the random experiment, and let X be the associated random variable. Thus, the probability density function (pdf), fx(x), has the value 0.0625 between -7 V and -3 V and the value 0.1875 between +3 V and +7 V and has the value 0 otherwise a. Sketch pdf fx(x) versus x. Draw properly labeled and scaled axes b. Directly beneath the plot of fx(x) in part a and to the same horizontal scale, sketch cumulative distribution function (CDF) Fx(x) versus x. Draw properly labeled and scaled axes Use unit step functions to express fx(x) as a single mathematical function for-oo < x Express Fx(x) mathematically for -00 < x <oo. You may express Fx(x) piecewise or using unit step functions c. d. (16) 2. Consider a random experiment with outcome x and associated random variable X with Gaussian probability density function fx(x). Let random variable X have expected value or mean μΧ--4.0 and standard deviation σχ 2.5. Define random variable Y g(x)-3x + 2, so that function y g(x) 3x2 a. Write the mathematical expression for fx(x b. Show the derivation of fy(y), the probability density function for random variable Y c. From the structure of fy(y), give the numerical values for the mean, Hy, and standard deviation, ơY, of random variable Y (17) 3. Consider a random experiment with numerical outcome x and associated random variable X with probability density function fx(x) 0.256(x-2) + 0.50 (x-3) + 0.256(x-4), mean x, and variance ơX2 a. By clear, step by step direct integration, find the numerical value of mean b. By clear, step by step mathematical derivation, find the numerical value of variance ơX You may use the results of part a

Answer 1

solution (2):

Le( y=g(x)= 3x42 fx(거 ) Lee y = 3x+ 2 3x 6 ": 56.2s