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統計検定 1級過去問解答/解答例と解説
2014年11月30日 (日) 試験

統計数理問2 [2]

設問の要約

平均と分散を求めよ．

解答例

M_{X}^{'} (t) = E [X e^{t X}] = m (1 - t)^{- m - 1}

$\begin{equation} M_X'(t) = E[Xe^{tX}] = m(1 - t)^{-m-1} \end{equation}$

E [X] = M_{X}^{'} (0) = m

$\begin{equation} E[X] = M_X'(0) = m \end{equation}$

M_{X}^{″} (t) = E [X^{2} e^{t X}] = m (m + 1) (1 - t)^{- m - 2}

$\begin{equation} M_X''(t) = E[X^2e^{tX}] = m(m + 1)(1 - t)^{-m-2} \end{equation}$

E [X^{2}] = M_{X}^{″} (0) = m (m + 1)

$\begin{equation} E[X^2] = M_X''(0) = m(m + 1) \end{equation}$

V a r [X] = E [X^{2}] - (E [X])^{2} = m (m + 1) - m^{2} = m

$\begin{equation} Var[X] = E[X^2] - (E[X])^2 = m(m + 1) - m^2 = m \end{equation}$

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