Prob and Stat Homework 10
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Prob and Stat
Homework 10
Problem 1
For the random sample (X1 , . . . , Xn ) i.i.d. following N (µ, σ2 ), summarize three important distributions about the sample mean, the sample variance, and the ratio between the sample mean and sample standard deviation.
Problem 2
Prove that the sample variance is an unbiased estimator for population variance. (You can not directly use the conclusion from χ2 distribution, as the original random sample might not be normal.)
Hint: Decompose Xi − by (Xi − µ) − ( − µ). Then the key question becomes computing E{ (Xi − µ)( − µ)}.
Problem 3
Let Xi be independent r.v. following N (i, (i + 1)2 ), i = 1, 2, 3. Use these three r.v.’s to construct a statistic with (1) χ3(2) distribution and (2) t2 distribution.
Problem 4
Suppose (X1 , . . . , Xn ) is an i.i.d. N (µ1 , σ1(2)) random sample, and (Y1 , . . . , Yn ) is an i.i.d. N (µ2 , σ2(2)) random sample. The two samples are independent. Then
(a) Find the distribution of Xn − Yn .
(b) Show that corr(Xi − Xn , Xj − Xn ) = − for i j, where corr(Z1 , Z2 ) is the correlation between r.v. Z1 and Z2 . (Hint: Start from the covariance.)
2021-12-25