Statistics and Probability
Scope
Statistics and probability tests distribution theory, estimation, hypothesis testing, regression, and stochastic simulation as applied to insurance data. Korean exam questions often require correct standard errors and interpretation rather than long computation.
Core rules
- Common severity models: exponential, gamma, lognormal, Pareto; frequency: Poisson, negative binomial.
- MLE maximizes the likelihood; the score has expectation zero and the information gives asymptotic variance.
- Confidence interval for a mean with known or large-sample SD: x̄ ± z_{α/2} σ/√n.
- Hypothesis tests control Type I error; the p-value is the probability of data at least as extreme under H0, not the probability that H0 is true.
- Regression assumptions include linearity, independence, homoscedasticity, and correct functional form.
- GLMs extend regression to non-normal responses; overdispersion in Poisson claims suggests a negative binomial or quasi-Poisson model.
Worked example
A sample of 100 claims has mean KRW 50 and sample standard deviation KRW 15. Construct a 95% confidence interval for the population mean.
Standard error = 15/√100 = 1.5. Using z = 1.96, the interval is 50 ± 1.96×1.5 = 50 ± 2.94, or [47.06, 52.94]. Reasoning: the sample mean is approximately normal by the central limit theorem; the interval covers the true mean in repeated samples with 95% confidence. The width falls as n rises and as variability falls. If the population SD were unknown and the sample small, a t critical value would replace 1.96.
Traps
Using the sample SD as the standard error without dividing by √n is the classic error. A 95% confidence interval is not a 95% probability statement about a fixed parameter. One-sided and two-sided critical values differ. Check independence: clustered or repeated claims often violate it and understate standard errors.