## Question

###### 3. [6 pts] Let Xi, . . . , Xn be a random sample from a...

3. [6 pts] Let Xi, . . . , Xn be a random sample from a distribution with variance σ2 < oo. Find cov(X,-x,x) for i 1,..,n.

## Answers

Let x_1, x_2 --- x_n are part of random sample. hence they are independent. E(x_i) = mu v(x_i) = sigma^2, i = 1(x)n let cov (x_i, x_j) = {0, i notequalto j x^bar Sigma^n_i = 1 x_p cov(x_i, x^bar) = cov (x_i, 1/n Sigma^n_i = 1 x_i) = cov [x_i, 1/n (x_1 + x_2 = --- x_i + ---- + x_n] = 1/n [cov (x_i, x_1) + cov (x_i, x_2) + --- + cov (x_i, x_i) + --- cov (x_i, x_n) = 1/n [0 + 0 + --- + sigma^2 + --- + 0] = sigma^2/n

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