Bagaimana saya menghitung

Misalkan dan Φ ( ⋅ ) adalah fungsi kerapatan dan fungsi distribusi dari distribusi normal standar.$\phi(\cdot)$$\Phi(\cdot)$

Bagaimana seseorang dapat menghitung integral:

$$\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$$

Notasi yang lebih konvensional adalah

$$y(\mu, \sigma) = \int\Phi\left(\frac{x-\mu}{\sigma}\right)\phi(x) dx = \Phi\left(\frac{-\mu}{\sqrt{1+\sigma^2}}\right).$$

Ini dapat ditemukan dengan membedakan integral terhadap dan σ , menghasilkan integral elementer yang dapat diekspresikan dalam bentuk tertutup:$\mu$$\sigma$

$$\frac{\partial y}{\partial \mu}(\mu, \sigma) = -\frac{1}{\sqrt{2 \pi } \sqrt{\sigma ^2+1}}e^{-\frac{1}{2}\frac{\mu ^2}{\sigma ^2+1}},$$

$$\frac{\partial y}{\partial \sigma}(\mu, \sigma) = \frac{\mu\sigma }{\sqrt{2 \pi } \left(\sigma ^2+1\right)^{3/2}}e^{-\frac{1}{2}\frac{\mu ^2}{\sigma ^2+1}}.$$

Sistem ini dapat diintegrasikan, dimulai dengan kondisi awal = ∫ Φ ( x ) φ ( x ) d x = 1 / 2 , untuk mendapatkan solusi yang diberikan (yang mudah diperiksa oleh diferensiasi).$y(0,1)$$\int\Phi(x)\phi(x)dx$$1/2$

$X$$Y$$X \sim N(a,b^2)$$Y$

$$P\{X \leq Y \mid Y = w\} = P\{X \leq w\} = \Phi\left(\frac{w-a}{b}\right).$$ $$P\{X \leq Y\} = \int_{-\infty}^\infty P\{X \leq Y \mid Y = w\}\phi(w)\,\mathrm dw = \int_{-\infty}^\infty \Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw.$$ $P\{X \leq Y\} = P\{X-Y \leq 0\}$$\Phi(\cdot)$$X-Y \sim N(a,b^2+1)$ $$\int_{-\infty}^\infty \Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw = \Phi\left(\frac{-a}{\sqrt{b^2+1}}\right)$$

$$\begin{align*} I(\gamma) & =\int_{-\infty}^{\infty}\Phi(\xi x+\gamma)\mathcal{N}(x|0,\sigma^{2})dx, \end{align*}$$ $\gamma=-\xi\mu$$I(\gamma)$$I(0)=0$$\gamma$ $$\begin{align*} \frac{dI}{d\gamma} & =\int_{-\infty}^{\infty}\mathcal{N}((\xi x+\gamma)|0,1)\mathcal{N}(x|0,\sigma^{2})dx\\ & =\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\xi x+\gamma\right)^{2}\right)\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(-\frac{x^{2}}{2\sigma^{2}}\right)dx. \end{align*}$$ $$\begin{align*} \left(\xi x+\gamma\right)^{2}+\frac{x^{2}}{\sigma^{2}} & =\underbrace{\left(\xi^{2}+\sigma^{-2}\right)}_{=a}x^{2}+\underbrace{-2\gamma\xi}_{=b}x+\underbrace{\gamma^{2}}_{=c} \\ &=a\left(x-\frac{b}{2a}\right)^{2}+\left(c-\frac{b^{2}}{4a}\right) \left(c-\frac{b^{2}}{4a}\right)\\ & =\gamma^{2}-\frac{4\gamma^{2}\xi^{2}}{4\left(\xi^{2}+\sigma^{-2}\right)}\\ &=\gamma^{2}\left(1-\frac{\xi^{2}}{\xi^{2}+\sigma^{-2}}\right)\\ &=\gamma^{2}\left(\frac{1}{1+\xi^{2}\sigma^{2}}\right) \end{align*}$$ $$\begin{align*} \frac{dI}{d\gamma} & =\frac{1}{2\pi\sigma}\exp\left(-\frac{1}{2}\left(c-\frac{b^{2}}{4a}\right)\right)\sqrt{\frac{2\pi}{a}}\int_{-\infty}^{\infty}\sqrt{\frac{a}{2\pi}}\exp\left(-\frac{1}{2}a\left(x-\frac{b}{2a}\right)^{2}\right)dx\\ & =\frac{1}{2\pi\sigma}\exp\left(-\frac{1}{2}\left(c-\frac{b^{2}}{4a}\right)\right)\sqrt{\frac{2\pi}{a}}\\ &=\frac{1}{\sqrt{2\pi\sigma^{2}a}}\exp\left(-\frac{1}{2}\left(c-\frac{b^{2}}{4a}\right)\right)\\ & =\frac{1}{\sqrt{2\pi\left(1+\sigma^{2}\xi^{2}\right)}}\exp\left(-\frac{1}{2}\frac{\gamma^{2}}{1+\xi^{2}\sigma^{2}}\right) \end{align*}$$

$$\begin{align*} I(\gamma) &=\int_{-\infty}^{\gamma}\frac{1}{\sqrt{2\pi\left(1+\sigma^{2}\xi^{2}\right)}}\exp\left(-\frac{1}{2}\frac{z^{2}}{1+\xi^{2}\sigma^{2}}\right)dz\\ &=\Phi\left(\frac{\gamma}{\sqrt{1+\xi^{2}\sigma^{2}}}\right) \end{align*}$$

yang menyiratkan

$$\begin{align*} \int_{-\infty}^{\infty}\Phi(\xi x)\mathcal{N}(x|\mu,\sigma^{2})dx &=I(\xi\mu)\\ &=\Phi\left(\frac{\xi\mu}{\sqrt{1+\xi^{2}\sigma^{2}}}\right). \end{align*}$$