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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

X1, X2, ..., X25 are independent random variables, each with mean 40 and variance 100. What are the mean and variance of the sample mean of these 25 variables?

The sample mean has mean 40 and variance 4. The expectation of the mean equals the common mean, and independence makes its variance the individual variance divided by n, which is 100 divided by 25, giving 4.

  1. AMean 40, variance 4Correct
  2. BMean 40, variance 100
  3. CMean 1000, variance 2500
  4. DMean 40, variance 20
  5. Mean 8, variance 4

Explanation

The mean of the sample mean equals the common mean, 40. Its variance is sigma^2/n = 100/25 = 4. Option 'variance 20' wrongly uses the standard deviation 10 times 2 or a similar slip; option 2500 is the variance of the sum, not the mean.

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