> [!definition] > > $ > S(n) = X_1 + X_2 + \cdots X_n > $ > A [[Set|set]] $X_1, X_2, \cdots, X_n$ of [[Probabilistic Independence|independent]] and identically distributed (drawn from the same [[Population|population]]/[[Probability Distribution|probability distribution]]) [[Random Variable|random variables]] constitute a random [[Sample|sample]] of size $n$ for that population. > > Suppose that each [[Expectation|expected value]] $\ev(X_j) = \mu$ and [[Variance|variance]] $\var{X_j} = \sigma^2(1 \le j \le n)$, then $\ev{(S(n))} = n\mu$ and $\var{S(N)} = n\sigma^2$. > [!theorem] > > If $X_1$, $X_2$, $\cdots$, $X_n$ are independent identically distributed random variables with a common [[Moment Generating Function|moment generating function]] $M(t)$, > $ > M_{S(n)}(t) = \paren{M(t)}^n > $ > > *Proof*. By induction. > > $M_{S(1)} = M(t)$. > > If $M_{S(n)}(t) = (M(t))^n$, then $M_{S(n + 1)}(t) = (M(t))^n \cdot M(t)$ (see [[Probabilistic Independence|independence]]).