Exam MAS-I Practice Question 31
Study guide: Exam MAS-I
Claim sizes X_1, \ldots, X_n are a random sample from a continuous distribution with CDF F, mean \mu and finite variance \sigma^2 > 0. Let \bar{X} be the sample mean, S^2 = \frac{1}{n-1}\sum (X_i - \bar{X})^2 the sample variance, and X_{(n)} the largest claim.
Determine which of the following statements are true.
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I. Because
E[S^2] = \sigma^2, the sample standard deviationSis an unbiased estimator of\sigma. -
II. Whatever the shape of
F,\text{Var}(\bar{X}) = \sigma^2/n. -
III. For every
x,P(X_{(n)} \le x) = F(x)^n. -
A) I only
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B) I and II only
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C) I and III only
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D) II and III only
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E) The answer is not given by (A), (B), (C), or (D)
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