m : median. P(X \le m) = 1/2.
X_i’s are i.i.d. X_{(i)} = Y_i.
\begin{align*}
Z_i &= I(X_i \le m) \sim
bernoulli~\bigg({1 \over 2}\bigg)
\\[15pt]
Z &= Z_1 + \cdots Z_n \sim
binominal~\bigg(n,{1 \over 2} \bigg)
\end{align*}
\begin{align*}
\therefore~
P(Y_1 \le m) &=
P(\text{m보다 작은 표본이 1개 이상})
\\[10pt]
&= P(Z \ge 1) = 1 - P(Z=0)
\\[10pt]
&= 1 - \binom{n}{0}
\bigg(\dfrac{1}{2}\bigg)^0
\bigg(\dfrac{1}{2}\bigg)^n
\\[15pt]
&= 1-\bigg(\dfrac{1}{2}\bigg)^n
\\[30pt]
P(Y_2 \le m) &= P(Z \ge 2) =
1 - P(Z=0) - P(Z=1)
\\[10pt]
&= 1 - \binom{n}{0}
\bigg(\dfrac{1}{2}\bigg)^0
\bigg(\dfrac{1}{2}\bigg)^n
- \binom{n}{0}
\bigg(\dfrac{1}{2}\bigg)^1
\bigg(\dfrac{1}{2}\bigg)^{n-1}
\\[15pt]
&= 1-\bigg(\dfrac{1}{2}\bigg)^n
-n~\bigg(\dfrac{1}{2}\bigg)^n
\\[30pt]
P(m \le Y_7) &= 1 - P(Y_7 < m)
\\[10pt]
&= 1-P(\text{n개 중 m보다 작은 표본 수 : 7개 이상})
\\[10pt]
&= 1 - P(Z \le 7) = P(Z < 7)
\end{align*}
공식
\begin{align*}
P(Y_k < m) &
= \sum_{x=k}^{n}~\binom{n}{x}
\bigg({1 \over 2}\bigg)^n
= P(Z \ge k)
\\[20pt]
P(Y_k < m) &=
\sum_{x=0}^{k-1}~\binom{n}{x}
\bigg({1 \over 2}\bigg)^n
= P(Z < k)
\\[20pt]
P(Y_a < m < Y_b) &=
\sum_{x=a}^{b-1}~\binom{n}{x}
\bigg({1 \over 2}\bigg)^n
\end{align*}
예시 1
\begin{align*}
&\text{eg. n=5}~~~P(Y_1 < m < Y_5)
\\[10pt]
&\sum_{i=1}^{5-1}~\binom{n}{x}
\bigg({1\over 2}\bigg)^5
= 0.9375
\\[20pt]
&\text{median's 93.75\% confidence
interval is }[A_1, A_5]
\end{align*}
예시 2
\begin{align*}
&\text{eg. n=5}~~~P(Y_2 < m < Y_4)
\\[10pt]
&\sum_{i=2}^{4-1}~\binom{n}{x}
\bigg({1\over 2}\bigg)^5
= 0.625
\\[20pt]
&\text{median's 62.5\% confidence
interval is }[A_2, A_4]
\end{align*}