Question

(Central Limit Theorem) An insurance company serves 10 large customers. The insurance claim placed by each customer in a year is uniformly distributed between 0 and 100. Assume that the insurance claims from different customers are independent. Use the central limit theorem to approximately compute the probability that the total insurance claim placed by all customers in a year exceeds 625. Let Φ(x) denote the cumulative distribution function of a standard normal distribution, i.e. a normal distribution with mean 0 and variance 1. Express your answer in terms of Φ, and then use software of your choice to evaluate it. (Hint: Compute the mean and standard deviation of the total insurance claim placed in a year. By the central limit theorem, the total insurance claim is approximately normally distributed. Why?)

Answer 1

We want to know the probability of total insurance claims. Using the central limit theorem ,we can use a normal distribution for approximately calculating probabilties of the sum of independent values.

X: Individual claims

E(X) = (a + b) / 2 = (0 + 100) / 2 = 50

= 50

V(X) = = 100*100/ 12 = 833.33

= 833.33

For the sum of variables we have the following distribution

We have n = 10

.........

We want to find the the sum exceeds 625

P(X > 625) = P( Z > 1.37)

= 1 - P(Z<1.37)

= 1 - Φ(1.37) .................Where Φ denotes the cumulative standard normal probabiltiy.

= 1- 0.91454 .............using normal probabilty tables or excel function 'normsdist(z-score)'