이항 분포
- n번 시도 할 때 x번 성공할 확률. 복원추출
\begin{align*}
P(X=x) &= \binom{n}{x}~p^x~
(1-p)^{1-x}~ I(x\in\{0, 1, \cdots, n\}) \\[15pt]
M_{X}(t) &=
\bigg\{(1-p)+pe^t\bigg\}^n \\[15pt]
\mu &= np \\[10pt]
\sigma &= np(1-p)
\end{align*}
확률함수 조건 확인
\text{이항정리 : }(a + b)^n = \displaystyle\sum_{k=0}^{n} \binom{n}{k} a^k\; b^{n-k}
- P(X = x) \ge 0 : 만족
\begin{align*}
&P(X = x)
= \displaystyle\binom{n}{x}
~p^x~(1-p)^{(1-x)}
~I(x \in \{0, 1, \cdots, n\})
\end{align*}
\\[30pt]
\begin{align*}
&\text{이항정리: } \displaystyle\binom{n}{x}
= \dfrac{n!}{x!(n-x)!} \ge 0
\\[20pt]
&\text{확률함수: }
p^x~(1-p)^{(1-x)} \ge 0
\\[20pt]
&\text{지시함수: }
I(x \in \{0, 1, \cdots, n\})
\ge 0
\end{align*}
- \displaystyle\sum P(X = x) = 1 : 만족
\begin{align*}
\displaystyle\sum P(X = x)
&= \displaystyle\sum \binom{n}{x}
~p^x~(1-p)^{(1-x)}
~I(x \in \{0, 1, \cdots, n\})
\\[20pt]
&= \displaystyle\sum_{x=0}^{n} \binom{n}{x}
~p^x~(1-p)^{1-x}
\\[20pt]
&= \{p + (1-p)\}^{n}
\\[10pt]
&= 1
\end{align*}
1차 적률
\begin{align*}
E(X) &= \sum_{x}~xP(X=x) = np
\\[15pt]
&=~\sum_{x=0}^{n}~~x \cdot
\binom{n}{x}~p^x~(1-p)^{n-x}
\\[15pt]
&=~\sum_{x=1}^{n}~~x \cdot
\dfrac{n}{x}~\binom{n-1}{x-1}
~p^x~(1-p)^{n-x}
\\[15pt]
&=~n~\sum_{x=1}^{n}
~\binom{n-1}{x-1}~p^x~(1-p)^{n-x}
\\[15pt]
&= ~n~\sum_{k=0}^{n_*}
~\binom{n_*}{k}~p^x~(1-p)^{n-x}
\\[15pt]
&= ~np~\sum_{k=0}^{n_*}
~\binom{n_*}{k}~p^{x-1}~(1-p)^{n-x}
\\[15pt]
&= ~np
\end{align*}
~n_* = n-1\\
~k = x-1
2차 적률과 분산
\begin{align*}
E(X^2-X) &= E\big( X(X-1) \big)
\\[20pt]
= \sum_{x}&~ x(x-1)~P(X=x)
\\[15pt]
= \sum_{x}&~ x(x-1)
~\binom
{\textcolor{red}{n}}
{\textcolor{red}{x}}
~p^x~(1-p)^{x-1}
\\[15pt]
= \sum_{x}&~ x(x-1) \cdot
\dfrac
{\textcolor{red}{n(n-1)}}
{\textcolor{red}{x(x-1)}} \cdot
\binom
{\textcolor{red}{n-2}}
{\textcolor{red}{x-2}}
~p^x\cdot(1-p)^{x-1}
\\[15pt]
=~n&(n-1)~\sum_{x}
\binom
{n-2}
{x-2}
~p^x~(1-x)^{n-x}
\\[15pt]
=~n&(n-1)~\sum_{x}
\binom
{n_*}
{k}
(1-p)^{n_* - k}
\\[15pt]
=~n&(n-1)~p^2
\end{align*}
기댓값: 제곱
\begin{align*}
E(X^2) &= E(X^2 - X + X)
= E(X^2 - X) + E(X)
\\[20pt]
&= n(n-1)~p^2 + np
\\[20pt]
&= n^2p^2 - np^2 + np
\end{align*}
분산
\begin{align*}
Var(X) &= E(X^2) - E(X)^2
\\[10pt]
&= E(X^2 - X) + E(X) - E(X)^2
\\[10pt]
&= n^2p^2 - np^2 + np - n^2p^2
\\[10pt]
&= np(1-p)
\end{align*}
3차 적률
- 정의
\begin{align*}
E(X^3) &= np~[(n-1)p+1]~[(n-2)p+1]
\\[20pt]
= E \bigg(& X(X-1)(X-2)
+ 3X^2 -3X + X\bigg)
\\[15pt]
= E \bigg(& X(X-1)(X-2)
+ 3X^2 -2X\bigg)
\\[15pt]
= E \bigg(& X(X-1)(X-2) \bigg)
+ 3~E \bigg( X(X-1) \bigg)
- E(X)
\end{align*}
- 중간과정: X(X-1)(X-2)
\begin{align*}
&E\bigg( X(X-1)(X-2) \bigg)
\\[20pt]
&=~\sum_{x}~x~(x-1)~(x-2)~
P(X=x)
\\[20pt]
&=~x~(x-1)~(x-2)
\cdot
~\dfrac
{n(n-1)(n-2)}
{x(x-1)(x-2)}
\cdot
\binom
{n-3}
{x-3}
~p^x~(1-p)^{1-x}
\\[20pt]
&=~n~(n-1)~(n-2)~p^3
\end{align*}
- 다시 계산
\begin{align*}
E(X^3) &= E \bigg( X(X-1)(X-2) \bigg) + 3~E \bigg( X(X-1) \bigg)
- E(X)
\\[20pt]
= E \bigg(& X(X-1)(X-2)
+ 3X^2 -3X + X\bigg)
\\[15pt]
= E \bigg(& X(X-1)(X-2)
+ 3X^2 -2X\bigg)
\\[15pt]
= E \bigg(& X(X-1)(X-2) \bigg)
+ 3~E \bigg( X(X-1) \bigg)
- E(X)
\\[15pt]
= n(n&-1)(n-2)~p^3 + 3n(n-1)p^2 + np
\\[15pt]
= np[&(n-1)(n-2)p^2 + 3(n-1)p + 1]
\\[15pt]
= np[&(n-1)p + 1][(n-2)p + 1]
\end{align*}
MGF
\begin{align*}
M_Y(t) &= \sum_{y}~e^{ty}~P(Y=y)
= \sum_{y}~e^{ty}~\binom{n}{y}~p^y~(1-p)^{n-y}
\\[20pt]
&= \sum_{y}~\binom{n}{y}~(pe^t)^y~(1-p)^{n-y}
\\[20pt]
M_Y(t) &= \{pe^t + 1-p\}^{n}
\end{align*}