Let X1, . . . , Xn ∼ N (μ, σ2) be i.i.d. where μ is known, but σ2 is unknown. Consider the estimate σ2 = 1/n sum of (xi-u)^2 . Show that the sum of ((xi-u)/o)^2 has a chi squared distribution with n degrees of freedom. For α ∈ (0,1), denote the α-quantile of χ2(n) by qn(α). Using this notation, find numbers an,bn ∈ R such that P(σ ̃2 < anσ2) = 2.5% and P(σ ̃2 > bnσ2) = 2.5%.

Q: 9. Show that exp(iz) = exp(iz) if and only if z = nл (n = 0, ±1, 2, ...). (Compare

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A: X is uniformly distributed over (0, 1) and Y is exponentially distributed with parameter .

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A: Given information: Given that the random variable X is symmetrically distributed with mean 0.

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A: We have to use squeeze theorem here.

Q: Find solution of the problem in the form of the normal mode u * eo-xlc) and discuss the result a²u…

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Q: Find the mean and standard deviation for each uniform continuous model U(1, 13) U(70, 200) U(3, 92)

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Q: Show that is Reciz) e -ImZ p (ii) Im(iz) = Relz). %3D

A: Solution:- we have to show that  (1)    Re(iZ) = - ImZ (2)    Im(iZ) = Re(Z) where , Z = x + iy

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A: Provided data are,

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A: According to the given information, X~U0,1 Y~exp3