東京工業大学 工学院 システム制御系 2022年度 問題3

Author

Miyake

Description

Kai

問1

\[ \begin{aligned} \sigma_S^2 &= E \left[ \left( S - E[S] \right)^2 \right] \\ &= E \left[ S^2 - 2E[S]S + \left( E[S] \right)^2 \right] \\ &= E \left[ S^2 \right] - 2 \left( E[S] \right)^2 + \left( E[S] \right)^2 \\ &= E \left[ S^2 \right] - \left( E[S] \right)^2 \end{aligned} \]

問2

\[ \begin{aligned} \frac{dM_X(t)}{dt} &= \lambda e^t \exp \left( \lambda (e^t-1) \right) \\ \frac{d^2 M_X(t)}{dt^2} &= \lambda e^t \left(\lambda e^t + 1 \right) \exp \left( \lambda (e^t-1) \right) \end{aligned} より、 \begin{aligned} \mu_X &= \frac{dM_X(0)}{dt} \\ &= \lambda \\ E \left[ X^2 \right] &= \frac{d^2 M_X(0)}{dt^2} \\ &= \lambda (\lambda + 1) \\ \sigma_X^2 &= E \left[ X^2 \right] - \mu_X^2 \\ &= \lambda \end{aligned} \]

を得る。

問3

\[ \begin{aligned} M_Y(t) &= \sum_{y=1}^\infty e^{yt} \cdot (1-q) q^{y-1} \\ &= \frac{1-q}{q} \sum_{y=1}^\infty \left( q e^t \right)^y \\ &= \frac{1-q}{q} \cdot q e^t \cdot \frac{1}{1-qe^t} \\ &= \frac{(1-q)e^t}{1-qe^t} \end{aligned} \]

問4

\(X,Y\) が独立なので \(E[XY]=E[X]E[Y]\) が成り立つことを考慮して、

\[ \begin{aligned} M_Z(t) &= E \left[ e^{tZ} \right] \\ &= E \left[ e^{t(X+Y)} \right] \\ &= E \left[ e^{tX} e^{tY} \right] \\ &= E \left[ e^{tX} \right] E \left[ e^{tY} \right] \\ &= M_X(t) M_Y(t) \end{aligned} \]

を得る。

問5

(1)

\[ \begin{aligned} M_Z(t) &= M_X(t) M_Y(t) \\ &= e^t \exp \left( e^t-1 \right) \cdot \frac{e^t}{2-e^t} \\ &= \frac{\exp \left( e^t-1 \right)}{2e^{-t}-1} \end{aligned} \]

(2)

\[ \begin{aligned} \frac{dM_Z(t)}{dt} &= \frac{\exp \left( e^t-1 \right) \left( 2 - e^t + 2e^{-t} \right)}{\left( 2e^{-t}-1 \right)^2} \end{aligned} \]

(3)

\[ \begin{aligned} \mu_Z &= \left. \frac{dM_Z(t)}{dt} \right|_{t=0} \\ &= 3 \end{aligned} \]