DISCRETE MATH.+ITS APPLICATIONS CUSTOM
8th Edition
ISBN: 9781307447118
Author: ROSEN
Publisher: MCG
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Chapter A, Problem 2E
To determine
To prove:
That for every nonzero real numberx, the multiplicative inverse ofxis unique.
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6.54 Let Y₁, Y2,..., Y, be independent Poisson random variables with means 1, 2,..., An
respectively. Find the
a probability function of
Y.
b conditional probability function of Y₁, given that Y = m.
Y₁ = m.
c conditional probability function of Y₁+Y2, given that
6.55 Customers arrive at a department store checkout counter according to a Poisson distribution
with a mean of 7 per hour. In a given two-hour period, what is the probability that 20 or more
customers will arrive at the counter?
6.56 The length of time necessary to tune up a car is exponentially distributed with a mean of
.5 hour. If two cars are waiting for a tune-up and the service times are independent, what is
the probability that the total time for the two tune-ups will exceed 1.5 hours? [Hint: Recall the
result of Example 6.12.]
6.57 Let Y, Y2,..., Y,, be independent random variables such that each Y, has a gamma distribution
with parameters a, and B. That is, the distributions of the Y's might have different a's, but…
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Chapter A Solutions
DISCRETE MATH.+ITS APPLICATIONS CUSTOM
Ch. A - Prove Theorem 3, which states that the...Ch. A - Prob. 2ECh. A - Prove that for all real numbers x and y,...Ch. A - Prob. 4ECh. A - Prob. 5ECh. A - Prob. 6ECh. A - Prob. 7ECh. A - Prob. 8ECh. A - Prob. 9ECh. A - Prob. 10E
Ch. A - Prove that for all real numbers w, x, y, and z, if...Ch. A - Prob. 12ECh. A - Prob. 13ECh. A - Prob. 14ECh. A - Prove that for all real numbers x, y, and z, if x...Ch. A - Prob. 16ECh. A - Prove that for all real numbers w, x, y, and z, if...Ch. A - Prove that for all positive real numbers x and y,...Ch. A - Prove that for every positive real number x, there...Ch. A - Prob. 20ECh. A - Define a relation on the set of ordered pairs of...Ch. A - Prob. 22E
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- 6.82 6.83 6.84 6.85 *6.86 6.87 If Y is a continuous random variable and m is the median of the distribution, then m is such that P(Ym) = P(Y ≥ m) = 1/2. If Y₁, Y2,..., Y, are independent, exponentially dis- tributed random variables with mean ẞ and median m, Example 6.17 implies that Y(n) = max(Y₁, Y., Y) does not have an exponential distribution. Use the general form of FY() (y) to show that P(Y(n) > m) = 1 - (.5)". Refer to Exercise 6.82. If Y₁, Y2,..., Y,, is a random sample from any continuous distribution with mean m, what is P(Y(n) > m)? Refer to Exercise 6.26. The Weibull density function is given by -my" m-le-y/a f(y)= α 0. y > 0, elsewhere, where a and m are positive constants. If a random sample of size n is taken from a Weibull distributed population, find the distribution function and density function for Y(1) = min(Y1, Y2,Y). Does Y(1) = have a Weibull distribution? Let Y₁ and Y2 be independent and uniformly distributed over the interval (0, 1). Find P(2Y(1) 0, elsewhere,…arrow_forwardWhich of the functions shown below is differentiable at = 0? Select the correct answer below: -7-6-5-4- -6-5-4-3-21, -7-6-5-4-3-2 -7-6-5-4-3-2-1 2 4 5 6 -1arrow_forwardRafael needs to make a total of 50 deliveries this week. So far he has completed 28 of them. What percentage of his total deliveries has Rafael completed?arrow_forward
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- 6.26 The Weibull density function is given by e-y/a f(y) = α 0. y > 0, elsewhere, where a and m are positive constants. This density function is often used as a model for the lengths of life of physical systems. Suppose Y has the Weibull density just given. Find a the density function of UY". b E(Y) for any positive integer k. 6.27 Let Y have an exponential distribution with mean ẞ. 6.28 6.29 a Prove that W = √Y has a Weibull density with α = ẞ and m = 2. b Use the result in Exercise 6.26(b) to give E(Yk/2) for any positive integer k. Let Y have a uniform (0, 1) distribution. Show that U = -2ln(Y) has an exponential distri- bution with mean 2. The speed of a molecule in a uniform gas at equilibrium is a random variable V whose density function is given by 6.30 6.31 6.32 f(v) = av²e-by², v > 0, where b = m/2kT and k, T, and m denote Boltzmann's constant, the absolute temperature, and the mass of the molecule, respectively. a Derive the distribution of W = mV2/2, the kinetic energy of…arrow_forwardcorrect answer is Acould you please show me how to compute using the residue theoremarrow_forwardthe correct answer is A please explainarrow_forward
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