= Exact bootstrap distribution for three observations
{title2=$\mathbb P_*(\overline X^*=7/3)=2/9$}
Take the original observations $(1,2,4)$, with <sample mean> $\overline x=7/3$. All $3^3=27$ ordered <bootstrap samples> have <probability> $1/27$. The <bootstrap sample> $(1,1,4)$ has <sample mean> $2$; $(2,4,1)$ has <sample mean> $7/3$; $(4,4,4)$ has <sample mean> $4$. These are illustrative possibilities, not three simulated measurements used to estimate the distribution.
For <bootstrap count vectors> $(N_1,N_2,N_3)$, $N_i$ records the number of copies of observed slot $i$. Their <multinomial distribution> is
$$
\mathbb P_*\bigl((N_1,N_2,N_3)=(a,b,c)\bigr)
=\frac{3!}{a!\,b!\,c!}\frac1{27},
\qquad a+b+c=3,
$$
and $\overline X^*=(N_1+2N_2+4N_3)/3$. Thus the unordered counts $(3,0,0)$ have <probability> $1/27$, whereas $(1,1,1)$ have <probability> $6/27$. <Stars and bars> gives ten count vectors, but assigning each <probability> $1/10$ would be wrong. Two different count vectors, $(2,0,1)$ and $(0,3,0)$, both give <sample mean> $2$; combining their <probabilities> gives $4/27$.
\Image[/bootstrap-three-point-exact-mean-distribution.png]
{title=Exact conditional <sampling distribution> of the <sample mean> from observations $(1,2,4)$}
{description=All 27 ordered <bootstrap samples> are enumerated. Bar labels give their counts out of 27, not empirical frequencies from a random simulation. The dashed line marks $\overline x=7/3$; the gaps reflect this discrete three-point <empirical measure>.}
The <conditional bootstrap variance of a sample mean> gives a compact check on the enumeration. Each resampled draw has <expected value> $7/3$ and <variance>
$$
\frac13\left[\left(1-\frac73\right)^2+\left(2-\frac73\right)^2+\left(4-\frac73\right)^2\right]=\frac{14}{9}.
$$
Averaging three <independent> draws divides that <variance> by three:
$$
\boxed{\mathbb E_*[\overline X^*]=\frac73,\qquad
\operatorname{Var}_*(\overline X^*)=\frac{14}{27}.}
$$
The exact conditional <bootstrap standard error> is $\sqrt{14/27}\simeq0.720$. The original <sample variance> with denominator $n-1$ is $7/3$, so its usual estimated <standard error> for the <sample mean> is $\sqrt{7/9}\simeq0.882$. Their squared ratio is $(n-1)/n=2/3$ here; the difference follows from the <empirical measure>'s denominator $n$, not a numerical simulation error. Neither number is the known true sampling <standard error> of the unknown source <probability distribution>.
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