Probability and Statistics (Spring 2013) Homework #2
1. In the following PDF’s, compute the constant B required for proper
normalization:
Cauchy (α < ∞, β > 0):
f (x) = B
1+[(x −α) / β]2. − ∞ < x < ∞ Maxwell (α > 0)
f (x) = Bx2e− x2/α2, x > 0
0, otherwise
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2. For these advanced PDF’s, compute the constant B required for proper
normalization:
Beta (b > −1, c > −1) : f (x) = Bxb(1− x)c, 0 ≤ x ≤ 1
0, otherwise
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Chi-square (σ > 0, n =1, 2,...) : f (x) = Bx(n/2)−1exp(−x / 2σ2), x > 0
0, otherwise
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3. When someone presses “SEND” on a cellular phone, the phone attempts to set up a call by transmitting a “SETUP” message to a nearby base station. The phone waits for a response and if none arrives tithing 0.5 seconds it tries again. If it doesn’t get a response after n = 6 tries the phone stops transmitting messages
and generates a busy signal.
(a) Draw a tree diagram that describes the call setup procedure.
(b) If all transmissions are independent and the probability is p that a “SETUP”
message will get thorough, what is the PMF of K, the number of messages transmitted in a call attempt?
(c) What is the probability that the phone will generate a busy signal?
(d) As manager of a cellular phone system, you want the probability of a busy signal to be less than 0.02 If p = 0.9, what is the minimum value of n necessary to achieve your goal?
4. The time-to-failure in months, X of the light bulbs produced at two manufacturing plants A and B obey, respectively, the following CDFs
FX(x) = (1− e− x/5)u(x) for plant A FX(x) = (1− e− x/2)u(x) for plant B.
Plant B produces two times as many bulbs as plant A. The bulbs indistinguishable to the eye, are intermingled and sold. What is the probability that a bulb purchased at random will burn at least two months?
5. The Sixers ant the Celtics play a best out of five playoff series. The series ends as soon as one of the teams has won three games. Assume that either team is
equally likely to win any game independently of any other game played. Find (a) The PMF PN(n) for the total number N of games played in the series;
(b) The PMF PW(n) for the number W of Celtic wins in the series;
(c) The PMF PL(n) for the number L of Celtic looses in the series;
6. The Zipf (n,α= 1) random variable X introduced in Problem 2.3.9 is often used to model the “Popularity” of collection of n objects. For example, a Web server can deliver one of n Web pages. The pages are numbered such that the page 1 is the most requested page, page 2 is the second most requested page, and so on. If
page k is requested, then X = k.
To reduce external network traffic, an ISP gateway caches copies of the k most popular pages. Calculate, as a function of n for1 ≤ n ≤ 1, 000, how large k must be to ensure that the cache can deliver a page with probability 0.75.
7. Let X be a continuous random variable with PDF f (x) = kx, 0 ≤ x ≤ 1
0, otherwise
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where k is a constant.
(a) Determine the value of k and sketch fX (x).
(b) Find and sketch the corresponding CDF FX (x).
(c) Find P (¼ < X ≤ 2)
8. The random variable X has CDF
FX(x) =
0 x < −3, 0.4 − 3 ≤ x < 5, 0.8 5 ≤ x < 7,
1 x ≥ 7.
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(a) Draw a graph of the CDF.
(b) Write PX (x), the PMF of X.
