Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the distribution...

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Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the distribution...

Let Y₁<Y₂<...<Y_{n be the
order statistics of a random sample of size n from the distribution
having p.d.f f(x) = e}^-y , 0<y< gif.latex?%5Cinfty , zero elsewhere. Answer the following questions.

(a) decide whether Z₁ = Y_2
andZ₂=Y₄-Y₂ are stochastically independent or not. (hint. first find the joint p.d.f. of Y_{2 and Y4)}

(b) show that

Z₁ = nY_1, Z₂= (n-1)(Y₂-Y₁), Z₃=(n-2)(Y₃-Y₂₎, ...., Z_n=Y_n-Y_n-1

are stocahstically independent and that each Z_i has the exponential distribution.(hint use change of variable technique)

math Statistics-And-Probability

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