IAI Actuarial Core Principles · Actuarial Statistics · Random sampling and sampling distributions
A random sample of n = 5 independent observations is drawn from a continuous distribution with cumulative distribution function F. What is the probability that the sample maximum is less than or equal to the median of the distribution?
The probability is 1/32. The maximum lies at or below the median only if all five independent observations do, and each does so with probability one half, so the probability is one half raised to the fifth power.
- A1/5
- B1/16
- C1/32Correct
- D1/10
- 1/2
Explanation
The maximum is at most the median only if every observation is at most the median. Each observation has probability 1/2 of doing so, so the probability is (1/2)^5 = 1/32. Option 1/16 would result from using n = 4, and 1/2 ignores the sample size.
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