![Consider a system consisting of three components as pictured. The system will continue to function as long as the first component functions and either component 2 or component 3 functions. Let X1, X2, and X3 denote the lifetimes of components 1, 2, and 3, respectively. Suppose the Xis are independent of one another and each X, has an exponential distribution with parameter λ. (a) Let Y denote the system lifetime. Obtain the cumulative distribution function of Y and differentiate to obtain the pdf. [Hint: F(y) P(Y S y); express the event Y S y} in terms of unions and/or intersections of the three events X1 y\, {X2 3 y), {X3 S yj.] cdf for y>o pdf for y>0 (b) Compute the expected system lifetime.](http://img.homeworklib.com/questions/1beeec70-fd1f-11eb-baf0-7de05de990ec.png?x-oss-process=image/resize,w_560)
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Consider a system consisting of three components as pictured. The system will continue to function as...
A machine needs 4 out of its 6 identical independent components to operate. Let X1, X2,...
A machine needs 4 out of its 6 identical independent components to operate. Let X1, X2, ..., X. denote the respective lifetimes of the components, and assume that each component's lifetime is exponentially distributed with a mean of 1/λ hours. Find: (a) The CDF of the machine's lifetime. (b) The PDF of the machine's lifetime.
A system contains two components X, Y which both need to work in order for the...
A system contains two components X, Y which both need to work in order for the system to run. The lifetime of component X is an exponential random variable X with parameter 2, and the lifetime of component Y is an exponential random variable Y with parameter 1. Assume that X,Y are independent. Let Z denote the lifetime of the system, which depends on X, Y a. Describe Z as a function of X, Y b. Find the PDF of...
5. Lec 17 function of pairs of R.V., 8 pts) Let X be the lifetime of a critical and expensive com...
5. Lec 17 function of pairs of R.V., 8 pts) Let X be the lifetime of a critical and expensive component in a system, which is exponentially distributed with mean 2 years. The system also has a cheaper backup component that can take over when the expensive component fails so that the system can provide continuous service while the more expensive system is being repaired. Let Y be the lifetime of the backup system, which is also exponentially distributed but...
Suppose a system of ive components Ai,1 Si S 5 is arranged as follows 2 Assum e the lifetime of each component is exponentially distributed with parameter) and the components function independently....
Suppose a system of ive components Ai,1 Si S 5 is arranged as follows 2 Assum e the lifetime of each component is exponentially distributed with parameter) and the components function independently. Let of the i-th component, that is the random variable defined by (Xi - t) means that the the i-th component stops working at time t. Saying that Xi has an exponenti distribution with parameter X means X, be the lifetime random variable and P(Xi s t)-1-e*. be...
14. Let X1, X2, X3 be independent random variables that represent lifetimes (in hours) of three...
14. Let X1, X2, X3 be independent random variables that represent lifetimes (in hours) of three key components of a device. Suppose their respective distributions are exponential with means 1000, 1500, and 2000. Let Y be the minimum of Xi, X2, X3 and compute P(Y 1000).
Suppose X = Exp(1) and Y= -ln(x) (a)Find the cumulative distribution function of Y . (b)...
Suppose X = Exp(1) and Y= -ln(x)
(a)Find the cumulative distribution function of Y .
(b) Find the probability density function of Y .
(c) Let X1, X2, ... , Xk be i.i.d. Exp(1), and let Mk =
max{X1,..... , Xk)(Maximum of X1, ..., Xk). Find the probability
density function of Mk.(Hint: P(min(X1, X2, X3) > k) = P(X1
>= k, X2 >= k, X3 >= kq, how about max ?)
(d) Show that as k → 00, the CDF...
Plz use MGF technique The lifetime of an electronic component in an HDTV is a random...
Plz use MGF technique
The lifetime of an electronic component in an HDTV is a random variable that can be modeled by the exponential distribution with a mean lifetime ß. Two components, X1 and X2, are randomly chosen and operated until failure. At that point, the lifetime of each component is observed. The mean lifetime of these two components is X1 + X2 X =- a) Find the probability density function of x using the MGF technique (the method of...
2. -30 a) The joint pdf of random variables X and Y is given by f(x,y)...
2. -30 a) The joint pdf of random variables X and Y is given by f(x,y) = 27ye-3 y<x<0, y >0. Show that the joint moment generating function(mgf) of X and Y is 27 M(t1, tz) = tı <3, tı + t, <3 (3 - tı) (3 - 7ı - t2) Use the joint mgf to obtain Cov(X,Y). b) Let X1, X2, X3 be independent random variables representing the lifetime of 3 electronic components with the following pdf, where X...
(a) Consider four independent rolls of a 6-sided die. Let X be the number of l's...
(a) Consider four independent rolls of a 6-sided die. Let X be the number of l's and let y be the number of 2's obtained. What is the joint PMF of X and Y? (b) Let X1, X2, X3 be independent random variables, uniformly distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is the middle of the three values). Find the conditional CDF of X1, given that Y = 0.5. Under this conditional distribution, is...
Q. 2 (Gamma and exponential, 30 pts). A parallel system consists of two components with independent...
Q. 2 (Gamma and exponential, 30 pts). A parallel system consists of two components with independent lifetimes. The lifetime Ly of the first component is memoryless: it has the exponential distribution with parameter 1. On the other hand, based on statistical analyses, it is found out that the lifetime L2 of the second component has two independent phases each of which has the same characteristics as L). Therefore, it is assumed that L2 has the gamma distribution with shape index...
5. Lec 17 function of pairs of R.V., 8 pts) Let X be the lifetime of a critical and expensive component in a system, which is exponentially distributed with mean 2 years. The system also has a cheaper backup component that can take over when the expensive component fails so that the system can provide continuous service while the more expensive system is being repaired. Let Y be the lifetime of the backup system, which is also exponentially distributed but...
Suppose a system of ive components Ai,1 Si S 5 is arranged as follows 2 Assum e the lifetime of each component is exponentially distributed with parameter) and the components function independently. Let of the i-th component, that is the random variable defined by (Xi - t) means that the the i-th component stops working at time t. Saying that Xi has an exponenti distribution with parameter X means X, be the lifetime random variable and P(Xi s t)-1-e*. be...
14. Let X1, X2, X3 be independent random variables that represent lifetimes (in hours) of three key components of a device. Suppose their respective distributions are exponential with means 1000, 1500, and 2000. Let Y be the minimum of Xi, X2, X3 and compute P(Y 1000).
Suppose X = Exp(1) and Y= -ln(x)
(a)Find the cumulative distribution function of Y .
(b) Find the probability density function of Y .
(c) Let X1, X2, ... , Xk be i.i.d. Exp(1), and let Mk =
max{X1,..... , Xk)(Maximum of X1, ..., Xk). Find the probability
density function of Mk.(Hint: P(min(X1, X2, X3) > k) = P(X1
>= k, X2 >= k, X3 >= kq, how about max ?)
(d) Show that as k → 00, the CDF...
Plz use MGF technique
The lifetime of an electronic component in an HDTV is a random variable that can be modeled by the exponential distribution with a mean lifetime ß. Two components, X1 and X2, are randomly chosen and operated until failure. At that point, the lifetime of each component is observed. The mean lifetime of these two components is X1 + X2 X =- a) Find the probability density function of x using the MGF technique (the method of...
2. -30 a) The joint pdf of random variables X and Y is given by f(x,y) = 27ye-3 y<x<0, y >0. Show that the joint moment generating function(mgf) of X and Y is 27 M(t1, tz) = tı <3, tı + t, <3 (3 - tı) (3 - 7ı - t2) Use the joint mgf to obtain Cov(X,Y). b) Let X1, X2, X3 be independent random variables representing the lifetime of 3 electronic components with the following pdf, where X...
(a) Consider four independent rolls of a 6-sided die. Let X be the number of l's and let y be the number of 2's obtained. What is the joint PMF of X and Y? (b) Let X1, X2, X3 be independent random variables, uniformly distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is the middle of the three values). Find the conditional CDF of X1, given that Y = 0.5. Under this conditional distribution, is...
Q. 2 (Gamma and exponential, 30 pts). A parallel system consists of two components with independent lifetimes. The lifetime Ly of the first component is memoryless: it has the exponential distribution with parameter 1. On the other hand, based on statistical analyses, it is found out that the lifetime L2 of the second component has two independent phases each of which has the same characteristics as L). Therefore, it is assumed that L2 has the gamma distribution with shape index...
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