Wektor losowy \(\displaystyle{ (X,Y)}\) ma rozkład jednostajny na kole \(\displaystyle{ B((x_0,y_0),r)}\) oblicz \(\displaystyle{ \EE (X|Y)}\).
ROZWIĄZANIE
\(\displaystyle{ f_{(X|Y)}(x,y) = \pi r^2 1{\hskip -2.5 pt}\hbox{l}_{B((x_0,y_0),r)} \\
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f_Y(y) = \int_{-\sqrt{r^2-y^2}}^{\sqrt{r^2-y^2}} \pi r^2 dx = 2 \pi r^2 \sqrt{r^2-y^2} \\
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\EE (X|Y) = \int_{-\sqrt{r^2-y^2}}^{\sqrt{r^2-y^2}} X \frac{\pi r^2}{2 \pi r^2 \sqrt{r^2-y^2}} dx = \begin{cases} \ \frac{\sqrt{r^2-y^2}}{2} , y \in (-r,r) \\ 0 ,\quad \quad \quad \quad y \notin (-r,r) \end{cases}}\)
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