Problem set

Problem set BY lu19920218 ECE 302 Problem Set 9 Fall 2013 The following problems have been selected from the course text. 4. 78 In a large collection of wires, the length of a wire is X, an exponential random variable with mean 511 cm. Each wire is cut to make rings of diameter 1 cm. Find the probability mass function for the number of complete rings produced by each length of wire. 4. 85 The exam grades in a certain class have a Gaussian pdf with mean m and standard deviation o.

Find the constants a and b so that the random variable Y = ax + has a Gaussian pdf with mean m and standard deviation o . 4. 86, 4. 87 Let X = U n where n is a positive integer and U is a uniform random variable in the unit interval. Find the cdf and pdf of X. Repeat for the case where U is uniform in the interval [-1, 1]. 4. 94 modified Let Y = a where X is uniformly distributed in the interval (-1/2, 1/2). a. Show that Y is a Cauchy random variable. b. Find the pdf of Z = IN . 4. 96 Find the pdf of X??” – In(l – U where U is a uniform random variable in (O, 1). . 9 modified Let X be a random variable with mean m. Compare the Chebyshev inequality and the exact probability for the event {IX – m I > c} as a function of c for the case where: a. X is a uniform random variable in the interval b]; b. X has pdf fX(x) = 2 exp(-alxl); c. X is a zero mean Gaussian random variable with variance 0 2 . 4. 100 Let X be the number of successes in n Bernoulli trials where the probability of success is p. Let Y = X/n be the average number of successes per trial.

Apply the Chebyshev inequality o the event {IY – pl > a}. What happens as n ??”+ m? 4. 102 a. Find the characteristic function of the random variable X uniformly distributed over b). b. Find the mean and variance of X by applying the moment theorem. 4. 105 modified a. Show that the characteristic function of a Gaussian random variable X with mean m and variance 0 2 is 22 OX (w) = eJmw-o 12 . the characteristic function of Y = ax + b, a = O, where X is a Gaussian random variable. Hint: use the definition of (w), namely, OY (w) = E[eJwY ].