Penurunan perubahan variabel dari fungsi kepadatan probabilitas?

Dalam pengenalan pola buku dan pembelajaran mesin (rumus 1.27), itu memberi

di manax=g(y),px(x)adalah pdf yang sesuai denganpy(y

$$p_y(y)=p_x(x) \left | \frac{d x}{d y} \right |=p_x(g(y)) | g'(y) |$$ $x=g(y)$$p_x(x)$$p_y(y)$ sehubungan dengan perubahan variabel.

Buku-buku mengatakan itu karena pengamatan jatuh dalam kisaran akan, untuk nilai-nilai kecil δ x , ditransformasikan menjadi kisaran ( y , y + δ y ) .$(x, x + \delta x)$$\delta x$$(y, y + \delta y)$

Bagaimana ini diturunkan secara formal?

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Pembaruan dari Dilip Sarwate

Hasilnya hanya berlaku jika adalah peningkatan atau penurunan fungsi monoton secara ketat.$g$

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Beberapa perubahan kecil pada jawaban LV Rao maka jika

$$\begin{equation} P(Y\le y) = P(g(X)\le y)= \begin{cases} P(X\le g^{-1}(y)), & \text{if}\ g \text{ is monotonically increasing} \\ P(X\ge g^{-1}(y)), & \text{if}\ g \text{ is monotonically decreasing} \end{cases} \end{equation}$$ $g$ secara monoton secara monoton meningkatkan f Y ( y ) = f X ( g - 1 ( y ) ) ⋅ $$F_{Y}(y)=F_{X}(g^{-1}(y))$$ jika secara monoton menurunkan FY(y)=1-FX(g-1(y))fY(y)=-fX(g-1(y))⋅ $$f_{Y}(y)= f_{X}(g^{-1}(y))\cdot \frac{d}{dy}g^{-1}(y)$$ $$F_{Y}(y)=1-F_{X}(g^{-1}(y))$$ $$f_{Y}(y)=- f_{X}(g^{-1}(y))\cdot \frac{d}{dy}g^{-1}(y)$$ $$\therefore f_{Y}(y) = f_{X}(g^{-1}(y)) \cdot \left | \frac{d}{dy}g^{-1}(y) \right |$$

Suppose $X$ is a continuous random variable with pdf f(x). If we define $Y=g(X)$, where g() is a monotone function, then the pdf of $Y$ is obtained as follows:

$$\begin{eqnarray*} P(Y\le y) &=& P(g(X)\le y)\\ &=& P(X\le g^{-1}(y))\\ or\;\;F_{Y}(y)&=& F_{X}(g^{-1}(y)),\quad \mbox{by the definition of CDF}\\ \end{eqnarray*}$$ By differentiating the CDFs on both sides w.r.t. $y$, we get the pdf of $Y$. The function g() could be either monotonically increasing or monotonically decreasing. If the function g() is monotonically increasing, then the pdf of $Y$ is given by $$\begin{equation*} f_{Y}(y)= f_{X}(g^{-1}(y))\cdot \frac{d}{dy}g^{-1}(y) \end{equation*}$$ and other hand, if it is monotonically decreasing, then the pdf of $Y$ is given by $$\begin{equation*} f_{Y}(y)= - f_{X}(g^{-1}(y))\cdot \frac{d}{dy}g^{-1}(y) \end{equation*}$$ The above two equations can be combined into a single equation: $$\begin{equation*} \therefore f_{Y}(y) = f_{X}(g^{-1}(y))\cdot |\frac{d}{dy}g^{-1}(y)| \end{equation*}$$