Problem #2 (5 points) The lifetimes of 6 components in an electronics device are known to be independent random variables governed by exponential distributions with means of 1000, 2000, 3000, 4000, 7000 and 10,000 hours, respectively. a) b) c) d) What is the probability that the lifetimes of all the components exceed 5000 hours? What are the mean and variance of life of all of the components? What is the probability that at least one component lifetime exceeds 10,000 hours? If the electronics device continues to operate only if all components operate, what is the mean life of the electronics device?
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You are given the following probability density function, 6x(r), for the cosine of the surface an...
You are given the following probability density function, φ2(x), for the cosine of the surface an...
You are given the following probability density function, φ2(x), for the cosine of the surface angle, X, of a laser etching tool. The distribution function has one parameter, α, and one constant, c. PX (x) =竺2-1 a) What is the value of the constant, c? b) What is the moment estimator for α? c) Explain how you can determine if this moment estimator is unbiased. d) Let S (ai... 2s) denote a random sample of sample size n-24 with sample...
2. A certain type of electronic component has a lifetime X (in hours) with probability density...
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The probability density function given below describes a probability distribution used to model scores on certain...
The probability density function given below describes a probability distribution used to model scores on certain exams/tests: ?(?)={(?+1)?? for 0≤?≤1, 0 otherwise. The parameter θ must be greater than 1. a. Find E(X). A random sample of 10 test-takers gives the following scores in proportions: 0.96 0.43 0.77 0.85 0.93 0.79 0.77 0.85 0.74 0.98 b. Using part a, find the method of moments estimator for θ using the first moment of X based on the data above. c. Find...
1.) Consider the following continuous probability density function with unknown population parameter 0. in the town...
1.) Consider the following continuous probability density function with unknown population parameter 0. in the town f(x) = S(0/39) x 0-1 LO posao se for 0<x<3 otherwise (a) (b) Demonstrate that ... f(x) dx = 1 (you may assume 0 > 0) Determine the moment estimator for 0 (based on a random sample of n observations).
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random var...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
Consider a random experiment that has as an outcome the number x. Let the associated variable be X, with true (population) and unknown probability density function fx(x), mean ux. and variance σχ2. A...
Consider a random experiment that has as an outcome the number x. Let the associated variable be X, with true (population) and unknown probability density function fx(x), mean ux. and variance σχ2. Assume that n-2 independent, repeated trials of the random experiment are performed, resulting in the 2-sample of numerical outcomes xi and x2 Let estimate μ X of true mean #xbe μχ = (x1+x2)/2. Then the random variable associated with estimate μ xis estimator random 1. a. Show the...
1. Consider a random experiment that has as an outcome the number x. Let the associated random variable be X, with true (population) and unknown probability density function fx(x), mean ux, and varia...
1. Consider a random experiment that has as an outcome the number x. Let the associated random variable be X, with true (population) and unknown probability density function fx(x), mean ux, and variance σχ2. Assume that n 2 independent, repeated trials of the random experiment are performed, resulting in the 2-sample of numerical outcomes x] and x2. Let estimate f x of true mean ux be μΧ-(X1 + x2)/2. Then the random variable associated with estimate Axis estimator Ax- (XI...
A certain type of electronic component has a lifetime Y (in hours) with probability density function given by
A certain type of electronic component has a lifetime Y (in hours) with probability density function given by That is, Y has a gamma distribution with parameters α = 2 and θ. Let denote the MLE of θ. Suppose that three such components, tested independently, had lifetimes of 120, 130, and 128 hours. a Find the MLE of θ. b Find E() and V(). c Suppose that actually equals 130. Give an approximate bound that you might expect for the error of estimation. d What...
Let with Y, Y, ..., Yn be i id random variables the following probability density function,...
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The probability density function of an exponentially distributed random variable with mean 1/λ is λe^−λt for t≥0. Suppose the lifetime of a particular brand of light bulb follows an exponential distribution with a mean of 1000 hours. If a light fixture is equipped with two such bulbs, then what is the probability that it still illuminates a room after 1000 hours? Develop your answer by evaluating a double integral. What assumption must you make about the respective lifetimes of the...
You are given the following probability density function, φ2(x), for the cosine of the surface angle, X, of a laser etching tool. The distribution function has one parameter, α, and one constant, c. PX (x) =竺2-1 a) What is the value of the constant, c? b) What is the moment estimator for α? c) Explain how you can determine if this moment estimator is unbiased. d) Let S (ai... 2s) denote a random sample of sample size n-24 with sample...
2. A certain type of electronic component has a lifetime X (in hours) with probability density function given by otherwise. where θ 0. Let X1, . . . , Xn denote a simple random sample of n such electrical components. . Find an expression for the MLE of θ as a function of X1 Denote this MLE by θ ·Determine the expected value and variance of θ. » What is the MLE for the variance of X? Show that θ...
1.) Consider the following continuous probability density function with unknown population parameter 0. in the town f(x) = S(0/39) x 0-1 LO posao se for 0<x<3 otherwise (a) (b) Demonstrate that ... f(x) dx = 1 (you may assume 0 > 0) Determine the moment estimator for 0 (based on a random sample of n observations).
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Consider a random experiment that has as an outcome the number x. Let the associated variable be X, with true (population) and unknown probability density function fx(x), mean ux. and variance σχ2. Assume that n-2 independent, repeated trials of the random experiment are performed, resulting in the 2-sample of numerical outcomes xi and x2 Let estimate μ X of true mean #xbe μχ = (x1+x2)/2. Then the random variable associated with estimate μ xis estimator random 1. a. Show the...
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