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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 the molecule. b Find E(W). A fluctuating electric current I may be considered a uniformly distributed random variable over the interval (9, 11). If this current flows through a 2-ohm resistor, find the probability density function of the power P = 21². The joint distribution for the length of life of two different types of components operating in a system was given in Exercise 5.18 by f(y1, y2) = (1/8)ye (1+2)/2, y₁ > 0, y2 > 0, elsewhere. The relative efficiency of the two types of components is measured by U = Y2/Y₁. Find the probability density function for U. In Exercise 6.5, we considered a random variable Y that has a uniform distribution on the interval [1, 5]. The cost of delay is given by U = 2Y2 +3. Use the method of transformations to derive the density function of U.

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 the molecule. b Find E(W). A fluctuating electric current I may be considered a uniformly distributed random variable over the interval (9, 11). If this current flows through a 2-ohm resistor, find the probability density function of the power P = 21². The joint distribution for the length of life of two different types of components operating in a system was given in Exercise 5.18 by f(y1, y2) = (1/8)ye (1+2)/2, y₁ > 0, y2 > 0, elsewhere. The relative efficiency of the two types of components is measured by U = Y2/Y₁. Find the probability density function for U. In Exercise 6.5, we considered a random variable Y that has a uniform distribution on the interval [1, 5]. The cost of delay is given by U = 2Y2 +3. Use the method of transformations to derive the density function of U.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
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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 the molecule. b Find E(W). A fluctuating electric current I may be considered a uniformly distributed random variable over the interval (9, 11). If this current flows through a 2-ohm resistor, find the probability density function of the power P = 21². The joint distribution for the length of life of two different types of components operating in a system was given in Exercise 5.18 by f(y1, y2) = (1/8)ye (1+2)/2, y₁ > 0, y2 > 0, elsewhere. The relative efficiency of the two types of components is measured by U = Y2/Y₁. Find the probability density function for U. In Exercise 6.5, we considered a random variable Y that has a uniform distribution on the interval [1, 5]. The cost of delay is given by U = 2Y2 +3. Use the method of transformations to derive the density function of U.
Transcribed Image Text: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 the molecule. b Find E(W). A fluctuating electric current I may be considered a uniformly distributed random variable over the interval (9, 11). If this current flows through a 2-ohm resistor, find the probability density function of the power P = 21². The joint distribution for the length of life of two different types of components operating in a system was given in Exercise 5.18 by f(y1, y2) = (1/8)ye (1+2)/2, y₁ > 0, y2 > 0, elsewhere. The relative efficiency of the two types of components is measured by U = Y2/Y₁. Find the probability density function for U. In Exercise 6.5, we considered a random variable Y that has a uniform distribution on the interval [1, 5]. The cost of delay is given by U = 2Y2 +3. Use the method of transformations to derive the density function of U.
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