第2講確率分布

ガンマ分布

$X_1,~X_2,~\ldots,~X_{\alpha}$ が独立に指数分布 $Exp(\lambda)$ に従うとき， $X_1 + X_2 + \cdots + X_{\alpha}$ は $Ga(\alpha, \lambda)$ に従う．

X \sim G a (α, λ)

$\begin{equation} X \sim Ga(\alpha, \lambda) \end{equation}$

α > 0

$\alpha > 0$ とする．

f (x) = {\begin{cases} \frac{λ^{α}}{Γ (α)} x^{α - 1} e^{- λ x} & (x \geq 0) \\ 0 & (x < 0) \end{cases}

$\begin{equation} f(x) = \begin{cases}\ds \frac{\lambda^{\alpha}}{\varGamma(\alpha)}x^{\alpha-1}e^{-\lambda x} & (x \ge 0) \\ 0 & (x < 0) \end{cases} \end{equation}$

E [X] = \frac{α}{λ}

$\begin{equation} E[X] = \frac{\alpha}{\lambda} \end{equation}$

E [X^{k}] = \frac{1}{λ^{k}} \frac{Γ (α + k)}{Γ (α)}

$\begin{equation} E[X^k] = \frac{1}{\lambda^{k}}\frac{\varGamma(\alpha+k)}{\varGamma(\alpha)} \end{equation}$

V [X] = \frac{α}{λ^{2}}

$\begin{equation} V[X] = \frac{\alpha}{\lambda^{2}} \end{equation}$

M_{X} (t) = {(\frac{λ}{λ - t})}^{α}

$\begin{equation} M_{X}(t) = \left(\frac{\lambda}{\lambda - t}\right)^{\alpha} \end{equation}$

M o d e [X] = \frac{α - 1}{λ}

$\begin{equation} Mode[X] = \frac{\alpha - 1}{\lambda} \end{equation}$

ガンマ関数

Γ (α) = \int_{0}^{\infty} x^{α - 1} e^{- x} d x

$\begin{equation} \varGamma(\alpha) = \int_{0}^{\infty}x^{\alpha-1}e^{-x}dx \end{equation}$

\frac{Γ (α)}{λ^{α}} = \int_{0}^{\infty} x^{α - 1} e^{- λ x} d x

$\begin{equation} \frac{\varGamma(\alpha)}{\lambda^{\alpha}} = \int_{0}^{\infty}x^{\alpha-1}e^{-\lambda x}dx \end{equation}$

Γ (n) = (n - 1)! (n \in N)

$\begin{equation} \varGamma(n) = (n-1)!~~~~(n \in \mathbb{N}) \end{equation}$

Γ (\frac{1}{2}) = \sqrt{π}

$\begin{equation} \varGamma\left(\frac{1}{2}\right) = \sqrt{\pi} \end{equation}$

再生性

$X \sim Ga(a, \lambda),~Y \sim Ga(b, \lambda)$ で独立のとき，

X + Y \sim G a (a + b, λ)

$\begin{equation} X + Y \sim Ga(a + b, \lambda) \end{equation}$

標準正規分布との関係

N (0, 1)^{2} \sim G a (\frac{1}{2}, \frac{1}{2})

$\begin{equation} N(0,1)^{2} \sim Ga\left(\frac{1}{2},\frac{1}{2}\right) \end{equation}$

ポアソン分布との関係

$W \sim Ga(k,\lambda)$: $k$ 個の製品が故障するまでの時間
$X \sim Po(\lambda\omega)$: $\omega$ の間に故障する製品の数

\begin{aligned} P (W > ω) & = \int_{ω}^{\infty} \frac{λ^{k}}{Γ (k)} t^{k - 1} \exp [- λ t] d t \\ = {[\frac{λ^{k}}{Γ (k)} t^{k - 1} \cdot (- \frac{1}{λ}) \exp [- λ t]]}_{ω}^{\infty} - \int_{ω}^{\infty} (k - 1) \frac{λ^{k}}{Γ (k)} t^{k - 2} \cdot (- \frac{1}{λ}) \exp [- λ t] d t \\ = \frac{(λ)^{k - 1}}{Γ (k)} ω^{k - 1} \cdot \exp [- λ ω] + \int_{ω}^{\infty} (k - 1) \frac{λ^{k - 1}}{Γ (k)} t^{k - 2} \cdot \exp [- λ t] d t \\ = \frac{(λ ω)^{k - 1}}{(k - 1)!} \exp [- λ ω] + \int_{ω}^{\infty} \frac{λ^{k - 1}}{Γ (k - 1)} t^{k - 2} \cdot \exp [- λ t] d t \\ = \frac{(λ ω)^{k - 1}}{(k - 1)!} \exp [- λ ω] + \frac{(λ ω)^{k - 2}}{(k - 2)!} \exp [- λ ω] + \int_{ω}^{\infty} \frac{λ^{k - 2}}{Γ (k - 2)} t^{k - 3} \cdot \exp [- λ t] d t \\ = \frac{(λ ω)^{k - 1}}{(k - 1)!} \exp [- λ ω] + \frac{(λ ω)^{k - 2}}{(k - 2)!} \exp [- λ ω] + \dots + \frac{(λ ω)^{1}}{(1)!} \exp [- λ ω] + \int_{ω}^{\infty} \frac{λ^{1}}{Γ (1)} t^{1 - 1} \cdot \exp [- λ t] d t \\ = \frac{(λ ω)^{k - 1}}{(k - 1)!} \exp [- λ ω] + \frac{(λ ω)^{k - 2}}{(k - 2)!} \exp [- λ ω] + \dots + \frac{(λ ω)^{1}}{(1)!} \exp [- λ ω] + \frac{λ^{1}}{Γ (1)} \int_{ω}^{\infty} \exp [- λ t] d t \\ = \frac{(λ ω)^{k - 1}}{(k - 1)!} \exp [- λ ω] + \frac{(λ ω)^{k - 2}}{(k - 2)!} \exp [- λ ω] + \dots + \frac{(λ ω)^{1}}{(1)!} \exp [- λ ω] + \frac{λ^{1}}{Γ (1)} {[- \frac{1}{λ} \exp [- λ t]]}_{ω}^{\infty} \\ = \frac{(λ ω)^{k - 1}}{(k - 1)!} \exp [- λ ω] + \frac{(λ ω)^{k - 2}}{(k - 2)!} \exp [- λ ω] + \dots + \frac{(λ ω)^{1}}{(1)!} \exp [- λ ω] + \frac{λ^{1}}{Γ (1)} \cdot \frac{1}{λ} \exp [- λ ω] \\ = \frac{(λ ω)^{k - 1}}{(k - 1)!} \exp [- λ ω] + \frac{(λ ω)^{k - 2}}{(k - 2)!} \exp [- λ ω] + \dots + \frac{(λ ω)^{1}}{(1)!} \exp [- λ ω] + \frac{(λ ω)^{0}}{0!} \exp [- λ ω] \\ = \sum_{x = 0}^{k - 1} \frac{(λ ω)^{x}}{x!} \exp [- λ ω] \\ = P (X \leq k - 1) \end{aligned}

$\begin{align} P(W > \omega) &= \int_{\omega}^{\infty}\frac{\lambda^{k}}{\varGamma(k)}t^{k-1}\exp[-\lambda t]dt \\ &=\left[\frac{\lambda^{k}}{\varGamma(k)}t^{k-1}\cdot\left(-\frac{1}{\lambda}\right)\exp[-\lambda t]\right]_{\omega}^{\infty} - \int_{\omega}^{\infty} (k-1)\frac{\lambda^{k}}{\varGamma(k)}t^{k-2}\cdot\left(-\frac{1}{\lambda}\right)\exp[-\lambda t]dt \\ &=\frac{(\lambda)^{k-1}}{\varGamma(k)}\omega^{k-1}\cdot\exp[-\lambda \omega] + \int_{\omega}^{\infty} (k-1)\frac{\lambda^{k-1}}{\varGamma(k)}t^{k-2}\cdot\exp[-\lambda t]dt \\ &=\frac{(\lambda\omega)^{k-1}}{(k-1)!}\exp[-\lambda \omega] + \int_{\omega}^{\infty} \frac{\lambda^{k-1}}{\varGamma(k-1)}t^{k-2}\cdot\exp[-\lambda t]dt \\ &=\frac{(\lambda\omega)^{k-1}}{(k-1)!}\exp[-\lambda \omega] + \frac{(\lambda\omega)^{k-2}}{(k-2)!}\exp[-\lambda \omega] + \int_{\omega}^{\infty} \frac{\lambda^{k-2}}{\varGamma(k-2)}t^{k-3}\cdot\exp[-\lambda t]dt \\ &= \frac{(\lambda\omega)^{k-1}}{(k-1)!}\exp[-\lambda \omega] + \frac{(\lambda\omega)^{k-2}}{(k-2)!}\exp[-\lambda \omega] + \cdots + \frac{(\lambda\omega)^{1}}{(1)!}\exp[-\lambda \omega] + \int_{\omega}^{\infty} \frac{\lambda^{1}}{\varGamma(1)}t^{1-1}\cdot\exp[-\lambda t]dt \\ &= \frac{(\lambda\omega)^{k-1}}{(k-1)!}\exp[-\lambda \omega] + \frac{(\lambda\omega)^{k-2}}{(k-2)!}\exp[-\lambda \omega] + \cdots + \frac{(\lambda\omega)^{1}}{(1)!}\exp[-\lambda \omega] + \frac{\lambda^{1}}{\varGamma(1)}\int_{\omega}^{\infty}\exp[-\lambda t]dt \\ &= \frac{(\lambda\omega)^{k-1}}{(k-1)!}\exp[-\lambda \omega] + \frac{(\lambda\omega)^{k-2}}{(k-2)!}\exp[-\lambda \omega] + \cdots + \frac{(\lambda\omega)^{1}}{(1)!}\exp[-\lambda \omega] + \frac{\lambda^{1}}{\varGamma(1)}\left[-\frac{1}{\lambda}\exp[-\lambda t]\right]_{\omega}^{\infty} \\ &= \frac{(\lambda\omega)^{k-1}}{(k-1)!}\exp[-\lambda \omega] + \frac{(\lambda\omega)^{k-2}}{(k-2)!}\exp[-\lambda \omega] + \cdots + \frac{(\lambda\omega)^{1}}{(1)!}\exp[-\lambda \omega] + \frac{\lambda^{1}}{\varGamma(1)}\cdot\frac{1}{\lambda}\exp[-\lambda \omega] \\ &= \frac{(\lambda\omega)^{k-1}}{(k-1)!}\exp[-\lambda \omega] + \frac{(\lambda\omega)^{k-2}}{(k-2)!}\exp[-\lambda \omega] + \cdots + \frac{(\lambda\omega)^{1}}{(1)!}\exp[-\lambda \omega] + \frac{(\lambda\omega)^{0}}{0!}\exp[-\lambda \omega] \\ &= \sum_{x=0}^{k-1}\frac{(\lambda\omega)^{x}}{x!}\exp[-\lambda\omega] \\ &= P(X \le k-1) \end{align}$

ガンマ分布を特殊化した分布

G a (1, λ) ⟹ E x p (λ)

$\begin{equation} Ga(1,\lambda)~~\Longrightarrow~~Exp(\lambda) \end{equation}$

G a (\frac{n}{2}, \frac{1}{2}) ⟹ χ^{2} (n)

$\begin{equation} Ga\left(\frac{n}{2},\frac{1}{2}\right)~~\Longrightarrow~~\chi^2(n) \end{equation}$

G a (n, λ) ⟹ E r (n, λ)

$\begin{equation} Ga\left(n, \lambda\right)~~\Longrightarrow~~Er(n, \lambda) \end{equation}$

補足1: $M_X(t)$ の計算

\begin{aligned} M_{x} (t) & = E [e^{X t}] \\ = \int_{0}^{\infty} e^{t x} \frac{λ^{α}}{Γ (α)} x^{α - 1} e^{- λ x} d x \\ = \frac{λ^{α}}{Γ (α)} \int_{0}^{\infty} x^{α - 1} e^{- (λ - t) x} d x \\ = \frac{λ^{α}}{Γ (α)} \frac{Γ (α)}{{(λ - t)}^{α}} \\ = {(\frac{λ}{λ - t})}^{α} \end{aligned}

$\begin{align} M_x(t) &= E[e^{Xt}] \\ &= \int_{0}^{\infty} e^{tx} \frac{\lambda^{\alpha}}{\varGamma\left(\alpha\right)}x^{\alpha-1}e^{-\lambda x}\,dx \\ &= \frac{\lambda^{\alpha}}{\varGamma\left(\alpha\right)}\int_{0}^{\infty} x^{\alpha-1}e^{-(\lambda - t) x}\,dx \\ &= \frac{\lambda^{\alpha}}{\varGamma\left(\alpha\right)} \frac{\varGamma\left(\alpha\right)}{\left(\lambda - t\right)^{\alpha}} \\ &= \left(\frac{\lambda}{\lambda - t}\right)^{\alpha} \end{align}$

補足2: $E[X^k]$ の計算

\begin{aligned} E [X^{k}] & = \int_{0}^{\infty} x^{k} \frac{λ^{α}}{Γ (α)} x^{α - 1} e^{- λ x} \\ = \frac{λ^{α}}{Γ (α)} \int_{0}^{\infty} x^{α + k - 1} e^{- λ x} \\ = \frac{λ^{α}}{Γ (α)} \frac{Γ (α + k)}{λ^{α + k}} \\ = \frac{1}{λ^{k}} \frac{Γ (α + k)}{Γ (α)} \end{aligned}

$\begin{align} E[X^k] &= \int_{0}^{\infty}x^{k}\frac{\lambda^\alpha}{\varGamma(\alpha)}x^{\alpha-1}e^{-\lambda x} \\ &= \frac{\lambda^\alpha}{\varGamma(\alpha)}\int_{0}^{\infty}x^{\alpha+k-1}e^{-\lambda x} \\ &= \frac{\lambda^\alpha}{\varGamma(\alpha)}\frac{\varGamma(\alpha+k)}{\lambda^{\alpha+k}} \\ &= \frac{1}{\lambda^{k}}\frac{\varGamma(\alpha+k)}{\varGamma(\alpha)} \end{align}$

第2講 確率分布