Consider n numbers x1, x2, . . . , xn laid out on a circle and...

Consider n numbers x1, x2, . . . , xn laid out on a circle and some value α. Consider the requirement that every number equals α times the sum of its two neighbors. For example, if α were zero, this would force all the numbers to be zero. (a) Show that, no matter what α is, the system has a solution. (b) Show that if α = 1 2 , then the system has a nontrivial solution. (c) Show that if α = − 1 2 , then there is a nontrivial solution if and only if n is even.

Expert Solution

Colby Messinger answered 1 year ago

Consider n numbers x1, x2, . . . , xn laid out on a circle and some value α.

Consider n numbers x1, x2, . . . , xn laid out on a circle and some value α. Consider the requirement that every number equals α times the sum of its two neighbors. For example, if α were zero, this would force all the numbers to be zero. (a) Show that, no matter what α is, the system has a solution. (b) Show that if α = 1/2 , then the system has a nontrivial solution. (c) Show...

Let X1, X2, . . . , Xn iid∼ N (µ, σ2 ). Consider the hypotheses...

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Let X1, X2, . . . , Xn be a random sample of size n from...

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Let X1, X2, · · · , Xn (n ≥ 30) be i.i.d observations from N(µ1,...

Let X1, X2, · · · , Xn (n ≥ 30) be i.i.d observations from N(µ1, σ12 ) and Y1, Y2, · · · , Yn be i.i.d observations from N(µ2, σ22 ). Also assume that X's and Y's are independent. Suppose that µ1, µ2, σ12 , σ22 are unknown. Find an approximate 95% confidence interval for µ1µ2.

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Consider the independent observations x1, x2, . . . , xn from the gamma distribution with pdf f(x) = (1/ Γ(α)β^α)x^(α−1)e ^(−x/β), x > 0 and 0 otherwise. a. Write out the likelihood function b. Write out a set of equations that give the maximum likelihood estimators of α and β. c. Assuming α is known, find the likelihood estimator Bˆ of β. d. Find the expected value and variance of Bˆ

Let X1, X2, X3, . . . be independently random variables such that Xn ∼ Bin(n,...

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Suppose we have a random sample of n observations {x1, x2, x3,…xn}. Consider the following estimator...

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Suppose X1, X2, . . . , Xn is a random sample from N(μ, 16). Find...

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Let x1 > 1 and xn+1 := 2−1/xn for n ∈ N. Show that xn is...

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Let X1,X2,...Xn be a random sample of size n form a uniform distribution on the interval...

Let X1,X2,...Xn be a random sample of size n form a uniform distribution on the interval [θ1,θ2]. Let Y = min (X1,X2,...,Xn). (a) Find the density function for Y. (Hint: find the cdf and then differentiate.) (b) Compute the expectation of Y. (c) Suppose θ1= 0. Use part (b) to give an unbiased estimator for θ2.

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