7. Suppose X₁ and X₂ are independent N(u, 1) random variables, for an unknown parameter u. (a) Find a sufficient statistic for μ. (b) Show that X² – 1 is an unbiased estimator of ². (c) Find a MVUE estimator of μ². You may use the fact that X₁|(X₁ + X₂ = y) ~ N(½, ½).

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A: Solution: Given information: n1=625   sample size of population 1n2= 625  sample size of population…

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A:

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A: Option (I) is correct as E(X2)>= (E(X))2 when we find Variance.

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Q: 7. Suppose that X is a random variable for which the m.g.f. is as follows: V(1) = -(3e' + e!) for -0…

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A: Given that  be a standard multi-variate Gaussian random variable.For are i.i.d. random variables.

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Q: 8. Let X1, X2 be i.i.d. random variables from a distribution with p.d.f xe r>0 fx(x) = otherwise and…

A: The distribution is given by, fX(x) = xe-xx>00Otherwise So,  P(X≥x)=∫x∞fX(t) dt=∫x∞te-t…

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A: i) First part:ii) second part:

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A: We are given joint PDF of X1 and X2 as,f(x1,x2)=12;for x1&gt;0 and x2&gt;0, x1+x2&lt;0…