The Normal and Related Distributions
This domain tests the normal distribution, the sampling distributions derived from it, and the central limit theorem that justifies their widespread use in actuarial work.
The normal distribution
- N(mu, sigma^2): symmetric, standardise with z = (x - mu)/sigma; roughly 68%, 95% and 99.7% of mass within 1, 2 and 3 standard deviations.
- Central limit theorem: the sum or mean of n independent identically distributed variables is approximately normal for large n, whatever the parent distribution. This underpins the normal approximation to binomial and Poisson counts and the asymptotic normality of maximum likelihood estimators.
Related distributions
- Lognormal: if Y = log X is normal then X is lognormal; used for claim sizes and, via geometric Brownian motion, for asset prices. E[X] = exp(mu + sigma^2/2); the median is exp(mu), always below the mean.
- Chi-square with k degrees of freedom: sum of k squared standard normals; used in variance tests and goodness-of-fit.
- Student t: Z divided by sqrt(chi-square_k / k); heavier tails, used when sigma is estimated from small samples.
- F: ratio of two scaled chi-squares; used for comparing variances and in ANOVA.
- QQ plots check normality far better than a single test statistic.
Worked example
Individual claims are iid with mean 100 and standard deviation 40 and n = 25. The sample mean has standard deviation 40/sqrt(25) = 8, not 40. So P(sample mean > 112) = P(Z > (112-100)/8) = P(Z > 1.5) = 0.0668. Beware: if claims are strongly skewed, n = 25 may be far too small for the approximation to bite, even though the algebra looks fine.
Traps that cost marks
- Using the standard deviation of X where the standard error sigma/sqrt(n) is required.
- Choosing t or z incorrectly when sigma is estimated.
- Quoting the lognormal mean where the median is the meaningful quantity.