Central Limit Theorem(s)

Definition:

• The sum of many independent RVs has an approximately normal distribution. Rule:
• 30 RVs is usually enough to justify a normal approximation for their sample mean (or for their sum).

Checking CLT with repeated samples using sample()

x = c(0, 0, 10, 50)
set.seed(1)
sample(x, 10, replace = TRUE) # randomly sample 10 times from x

Practice Problem

• Q: Suppose that each of 1,000 members of an insurance pool independently has a 0.6 probability of submitting a claim in a year. Using CLT, what is the approximate probability of receiving more than 630 claims?
• A:
  • Y = total number of claims submitted
    • This is a binomial RV with n = 1000, p = 0.6
  • E(Y) = n * p = 1000 * 0.6 = 600
  • var(Y) = np(1 - p) = 1000 * 0.6 * 0.4 = 240
  • sd(Y) = var(Y)^0.5 = 15.491

1 - pnorm(630, 600, 15.491) 

= 0.02639662