IAI Actuarial Core Principles · Actuarial Statistics · Random sampling and sampling distributions
Let X1, ..., X4 be independent Uniform(0, 1) random variables with order statistics X(1) < X(2) < X(3) < X(4). What is E[X(3)]?
E[X(3)] is 0.60. The k-th order statistic of n uniform observations follows a Beta(k, n-k+1) distribution with mean k/(n+1), so with k = 3 and n = 4 the expectation is 3/5.
- A0.50
- B0.60Correct
- C0.75
- D0.80
- 0.667
Explanation
For a Uniform(0,1) sample of size n, X(k) has a Beta(k, n-k+1) distribution with mean k/(n+1). Here k = 3 and n = 4, giving 3/5 = 0.60. The value 0.75 wrongly uses k/n.
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