= Three Bernoulli variables through characteristic functions
{c}
{title2=$\varphi_S(t)=(2/3+e^{it}/3)^3$}
Let $B_1,B_2,B_3$ be <independent random variables>, each with the <Bernoulli distribution> of parameter $1/3$, and let $S=B_1+B_2+B_3$. Multiplying their <characteristic functions> and expanding by the <binomial theorem> gives
$$
\varphi_S(t)=\left(\frac23+\frac13e^{it}\right)^3
=\frac8{27}+\frac{12}{27}e^{it}+\frac6{27}e^{2it}+\frac1{27}e^{3it}.
$$
The coefficients are exactly the <probability mass function> of the <binomial distribution> with parameters $3,1/3$: the sum takes $0,1,2,3$ with <probabilities> $8/27,12/27,6/27,1/27$. Direct outcome counting verifies them: an outcome with $k$ ones has <probability> $(1/3)^k(2/3)^{3-k}$, and there are $\binom3k$ such outcomes. The coefficients sum to one, recovering $\varphi_S(0)=1$.
At $t=\pi$, $\varphi_S(\pi)=(1/3)^3=1/27$. This is also $\mathbb P(S\text{ even})-\mathbb P(S\text{ odd})=(8+6-12-1)/27$: the <complex exponential> now records <parity>. The <moments> agree by two routes: <linearity of expectation> gives $\mathbb E[S]=1$, while the <variance of a sum> of <independent random variables> gives $\operatorname{Var}(S)=3(2/9)=2/3$. Summing $k^2\mathbb P(S=k)$ gives the <second moment> $5/3$, also recovered as $-\varphi_S''(0)$.
\Image[/characteristic-function-independent-sums-and-scaling.png]
{title=Independent products and the different limits of averages and variance-normalized sign sums}
{description=Top: exact <real parts> and <imaginary parts> of $\varphi_B(t)=2/3+e^{it}/3$ and its cube, the <characteristic function> of the three-variable <binomial distribution>. Bottom: for <independent> symmetric signs, $\cos(t/n)^n$ approaches one under averaging, whereas $\cos(t/\sqrt n)^n$ approaches $e^{-t^2/2}$ under <variance normalization>. All curves are exact model formulas; the finite curves illustrate the two limits rather than proving a general <central limit theorem>.}
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