과제3

M_{Y_i}(t) = (1-\beta t)^{-\alpha_i}

\begin{align*} &\text{if } X_i \text{들이 전부 독립인 경우,} \\[8pt] & M_{X_1 + X_2 + \cdots + X_n}(t) = M_{X_1}(t)M_{X_2}(t) \cdots M_{X_n}(t) \\[15pt] &=(1-\beta t)^{-\alpha_1}~ (1-\beta t)^{-\alpha_2}~ (1-\beta t)^{-\alpha_3} \cdots \\[15pt] &= (1-\beta t)^{-\sum\alpha_i} \end{align*}

\therefore~\sum_{i=1}^{n}~Y_i \sim Gamma\Big(\sum_{i=1}^{n}\alpha_{i,} ~\beta\Big)

\begin{align*} M_{kY}(t) &= E(e^{t\cdot kY}) = E(e^{tk \cdot Y}) = M_Y(tk) \\[10pt] M_{Y_i}(t) &= (1-\beta t)^{-\alpha_i} \\[10pt] M_{kY_i}(t) &= (1-\beta \cdot kt)^{-\alpha_i} = (1- k \beta \cdot t)^{-\alpha_i} \end{align*}

\therefore~kYi \sim Gamma(\alpha_{i,}~k \beta)

\begin{align*} E(S^2) &= E \bigg[{\sum_{i=1}^{n}(X_i - \overline{X})^2 \over n-1}\bigg] \\[15pt] &= {1 \over n-1}~ E \bigg[\sum_{i=1}^{n} ({X_i}^{2} + \overline{X}^2 -2X_i~\overline{X}) \bigg] \\[15pt] &= {1 \over n-1}~ E \bigg[\sum_{i=1}^{n} {X_i}^{2} + n\overline{X}^2 -2n~\overline{X}^2 \bigg] \\[15pt] &= {1 \over n-1}~ E \bigg[\sum_{i=1}^{n} {X_i}^{2} - n\overline{X}^2 \bigg] \\[15pt] &= {1 \over n-1}~ \bigg[ \sum_{i=1}^{n} E({X_i}^{2}) - nE(\overline{X}^2) \bigg] \\[15pt] &= {1 \over n-1}~ \Big[ nE({X_i}^{2}) - nE(\overline{X}^2)\Big] \\[15pt] &= {n \over n-1}\cdot [E({X_i}^{2})-E(\overline{X}^2)] \\[15pt] \end{align*}

\begin{align*} E(X^2) &= Var(X) + E(X)^2 = \sigma^2 + \mu^2 \\[15pt] E(\overline{X}^2) &= Var(\overline{X}) + E(\overline{X})^2 = {\sigma^2 \over n} + \mu^2 \end{align*}

E(S^2)= {n \over n-1}\cdot \bigg(\sigma^2 + \mu^2 - {\sigma^2 \over n} - \mu^2\bigg) = \sigma^2

\begin{align*} E(S^2) &= E \bigg[{\sum_{i=1}^{n}(X_i - \overline{X})^2 \over n-1}\bigg] \\[15pt] &= {1 \over n-1}\cdot E \bigg[\sum_{i=1}^{n}( {X_i}^{2} + \overline{X}^2 -2X_i~\overline{X}) \bigg] \\[15pt] &= {1 \over n-1}\cdot (n\lambda + n\lambda^2 - \lambda - n\lambda^2) \\[15pt] &= {1 \over n-1} \cdot (n-1)\lambda = \lambda \end{align*}

E(X) = np, \kern{10pt} Var(X)=np(1-p) \\[10pt] E(X^2) = np(1-p) + n^2p^2 \\[20pt] E(\overline{X}) = np, \kern{10pt} Var(\overline{X})={np(1-p) \over n} \\[10pt] E(\overline{X}^2)={np(1-p) \over n} + n^2p^2

\begin{align*} &E(S^2) = E \bigg[{\sum_{i=1}^{n}(X_i - \overline{X})^2 \over n-1}\bigg] \\[15pt] &= {n \over n-1}\cdot [E({X_i}^{2})-E(\overline{X}^2)] \\[15pt] &= {n \over n-1}\cdot \bigg[~np(1-p) + n^2p^2 - {np(1-p)\over n}- n^2p^2\bigg] \\[15pt] &= np(1-p)\end{align*}