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

5π 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 σ. Find the constants a and b so that the random variable Y = aX + b has a Gaussian pdf with mean m and standard deviation σ .

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 = α tan(πX), 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 = 1/Y .

4.96 Find the pdf of X = − ln(1 − U ), where U is a uniform random variable in (0, 1).

4.99 modiﬁed Let X be a random variable with mean m. Compare the Chebyshev inequality and the exact probability for the event {|X − m| > c} as a function of c for the case where:

a. X is a uniform random variable in the interval [−b, b];

b. X has pdf fX (x) =

α

2

exp(−α|x|);

c. X is a zero mean Gaussian random variable with variance σ 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 to the event

{|Y − p| > a}. What happens as n → ∞?

4.102

a. Find the characteristic function of the random variable X uniformly distributed over [−b, 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 σ 2 is

2 2

ΦX (ω) = ejmω−σ ω /2 .

b. Use the moment