# Distribution of Y in a Random Sample

Let X1, X2,...,Xn denote a random sample from a distribution that is N(μ, θ), 0<θ<infinity, where μ is unknown. Let Y = the sum from 1 to n of (Xi-Xbar)^2/n=V. What are the E[Y] and Var[Y].>0.

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Let X1, X2,..,Xn denote a random sample from a distribution that is N(μ, θ), 0< θ <infinity, where μ is ...

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Rayleigh density

1. A density function sometimes used by engineers to model length of life of electronic components is the Rayleigh density, given by if 0 < y1 <infinity, 0 otherwise. Assume Y1, Y2, ...Yn is a random sample from a Rayleigh distribution.

a. If Y has the Rayleigh density, find the probability density for U = Y2.

b. Use part a to find E(Y) and V(Y).

c. Use the first moment E(Y) to find a method of moment estimator for .

d. Use the second moment E(Y) to find a method of moment estimator for .

e. Find the MLE of . Denote this as .

f. What is the asymptotic variance of ?

g. Show whether is an unbiased estimator for .

h. Show whether is a consistent estimator for .

i. Show whether is sufficient for .