第2講確率分布

対数正規分布

$Y \sim N(\mu,\sigma^{2})$ のとき， $X=\exp(Y)$ が従う確率分布．

y = \log x

$\begin{equation} y = \log x \end{equation}$

\frac{d y}{d x} = \frac{1}{x}

$\begin{equation} \frac{dy}{dx} = \frac{1}{x} \end{equation}$

であるので，

f (x; μ, σ^{2}) = \frac{1}{\sqrt{2 π σ^{2}}} \frac{1}{x} \exp [- \frac{1}{2 σ^{2}} (\log x - μ)^{2}] (x > 0)

$\begin{equation} f(x; \mu, \sigma^{2}) = \frac{1}{\ds\sqrt{2\pi\sigma^{2}}}\frac{1}{x}\exp\left[-\frac{1}{2\sigma^{2}}(\log x - \mu)^{2}\right]~~~~(x > 0) \end{equation}$

E [X] = \exp [μ + \frac{σ^{2}}{2}]

$\begin{equation} E[X] = \exp\left[\mu + \frac{\sigma^{2}}{2}\right] \end{equation}$

V [X] = \exp [2 μ + σ^{2}] {\exp [σ^{2}] - 1}

$\begin{equation} V[X] = \exp\left[2\mu+\sigma^{2}\right]\left\{\exp\left[\sigma^{2}\right]-1\right\} \end{equation}$

補足1: $E[X]$ の計算

E [X] = E [e^{Y}] = M_{Y} (1)

$\begin{equation} E[X] = E[e^Y] = M_{Y}(1) \end{equation}$

ここで，

M_{Y} (t) = \exp [μ t + \frac{σ^{2}}{2} t^{2}]

$\begin{equation} M_{Y}(t) = \exp\left[\mu t + \frac{\sigma^2}{2}t^2\right] \end{equation}$

であるので，

E [X] = \exp [μ + \frac{σ^{2}}{2}]

$\begin{equation} E[X] = \exp\left[\mu + \frac{\sigma^2}{2}\right] \end{equation}$

補足2: $V[X]$ の計算

E [X^{2}] = E [(e^{Y})^{2}] = E [e^{2 Y}]

$\begin{equation} E[X^2] = E[(e^Y)^2] = E[e^{2Y}] \end{equation}$

ここで，

E [2 Y] = 2 μ

$\begin{equation} E[2Y] = 2\mu \end{equation}$

V [2 Y] = 4 σ^{2}

$\begin{equation} V[2Y] = 4\sigma^2 \end{equation}$

なので，

M_{2 Y} (t) = E [e^{2 Y t}] = \exp [2 μ t + \frac{4 σ^{2}}{2} t^{2}]

$\begin{equation} M_{2Y}(t) = E[e^{2Yt}] = \exp\left[2\mu t + \frac{4\sigma^2}{2}t^2\right] \end{equation}$

M_{2 Y} (1) = E [e^{2 Y}] = \exp [2 μ + 2 σ^{2}]

$\begin{equation} M_{2Y}(1) = E[e^{2Y}] = \exp\left[2\mu + 2\sigma^2\right] \end{equation}$

よって，

\begin{aligned} V [X] & = E [X^{2}] - (E [X])^{2} \\ = \exp [2 μ + 2 σ^{2}] - {(\exp [μ + \frac{σ^{2}}{2}])}^{2} \\ = \exp [2 μ + 2 σ^{2}] - \exp [2 μ + σ^{2}] \\ = \exp [2 μ + σ^{2}] {\exp [σ^{2}] - 1} \end{aligned}

$\begin{align} V[X] &= E[X^2] - (E[X])^2 \\ &= \exp\left[2\mu + 2\sigma^2\right] - \left(\exp\left[\mu + \frac{\sigma^2}{2}\right]\right)^2 \\ &= \exp\left[2\mu + 2\sigma^2\right] - \exp\left[2\mu + \sigma^2\right] \\ &= \exp\left[2\mu + \sigma^2\right]\left\{\exp[\sigma^2] - 1\right\} \\ \end{align}$

第2講 確率分布

対数正規分布

補足1: E[X]E[X] の計算

補足2: V[X]V[X] の計算

第2講確率分布

補足1: $E[X]$ の計算

補足2: $V[X]$ の計算