Independent and Identically Distributed

Study guide: Exam P-1

Random variables $X_1, X_2, \ldots, X_n$ are Independent and Identically Distributed (i.i.d.) when they are mutually independent and share one common distribution — the same CDF $F$ , hence the same mean $\mu$ and variance $\sigma^2$ . It is the standard model for a Random Sample and for a book of similar, unrelated policies.

$f(x_1, \ldots, x_n) = \prod_{i=1}^{n} f(x_i)$

$\begin{aligned} E[S_n] &= n\mu \\ \text{Var}(S_n) &= n\sigma^2 \end{aligned}$

Here $f$ is the common PMF or PDF and $S_n = X_1 + \cdots + X_n$ . The product form is the independence; using the same $f$ in every factor is the identical distribution. Each half can fail on its own.
Identically distributed is not identical. $X_1 + X_2$ has variance $2\sigma^2$ , but $2X_1$ has variance $4\sigma^2$ . Fifty independent policies are far less risky than one policy fifty times the size.
The mean of $S_n$ grows like $n$ but its standard deviation only like $\sqrt{n}$ , so the Coefficient of Variation of the total falls as $1/\sqrt{n}$ — the pooling effect behind the Law of Large Numbers. For the average, $E[\bar{X}] = \mu$ and $\text{Var}(\bar{X}) = \sigma^2/n$ .
The version of the Central Limit Theorem on Exam P assumes i.i.d. summands with finite variance; it is what licenses the Normal Approximation $S_n \approx N(n\mu,\ n\sigma^2)$ .
The Order Statistics formulas rely on it too: the maximum has $P(X_{(n)} \le x) = F(x)^n$ and the minimum has $P(X_{(1)} > x) = [1 - F(x)]^n$ .
Some i.i.d. sums are known exactly: Bernoulli $(p)$ terms sum to a binomial $(n, p)$ , Poisson $(\lambda)$ terms to a Poisson $(n\lambda)$ , and exponentials with mean $\theta$ to a Gamma $(n, \theta)$ .

Fifty Policies versus One Large Policy Each of 50 independent policies has an annual loss with mean $1,000 and standard deviation $400. Compare the total $S = X_1 + \cdots + X_{50}$ with a single risk $T = 50X_1$ that has the same expected loss.
Answer Both have mean $50 \times 1{,}000 = 50{,}000$ . The variances differ because the i.i.d. terms add variances, while scaling one risk squares the multiplier:
 $\begin{align*} \text{Var}(S) &= 50(400^2) \\ &= 8{,}000{,}000 \\ \text{SD}(S) &= 2{,}828 \\ \text{Var}(T) &= 50^2(400^2) \\ &= 400{,}000{,}000 \\ \text{SD}(T) &= 20{,}000 \end{align*}$ 
The coefficient of variation is $2{,}828/50{,}000 = 0.057$ for the pooled book against $0.40$ for the single large risk. Same expected loss, but pooling cuts the standard deviation to $1/\sqrt{50} \approx 14\%$ of the concentrated risk's.

The Largest of Five Claims Five claims are i.i.d. exponential with mean $2,000. Find the probability that the largest exceeds $5,000.
Answer One claim stays at or below $5,000 with probability $F(5{,}000) = 1 - e^{-2.5} = 0.917915$ . The maximum is at or below $5,000 only if all five claims are, and independence with a common $F$ turns that into a power:
 $\begin{align*} P(X_{(5)} > 5{,}000) &= 1 - F(5{,}000)^5 \\ &= 1 - 0.917915^5 \\ &= 1 - 0.651647 \\ &= 0.3484 \end{align*}$ 
A single claim exceeds $5,000 only $8.2\%$ of the time, but with five claims there is a $34.8\%$ chance at least one does — the reason a per-claim retention is hit far more often than a single claim's tail suggests.