球坐标方差

使用方差方程我们得到： $$\begin{equation*} Var(z) = E[z^2] - E[z]^2 \end{equation*}$$ 我们知道单位球面的方程为： $$\begin{equation*} x^2+y^2+z^2=1 \end{equation*}$$ 由于坐标是统一且独立选择的，因此 $z^2$ 组件将占据其中的大约三分之一。因此： $$\begin{equation*} E[z^2] = \frac13 \end{equation*}$$ 为了找到 $E[z]$，我们可以想象单位球体。所有坐标的平均值将位于 $(0, 0, 0)$ 的正中心，这就是我们期望的 z 值。因此： $$\begin{equation*} Var(z) = \frac13 - 0^2 = \boxed{\frac13} \end{equation*}$$

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Original Explanation

Using the variance equation we get: $$\begin{equation*} Var(z) = E[z^2] - E[z]^2 \end{equation*}$$

We know the equation of a the unit sphere is: $$\begin{equation*} x^2+y^2+z^2=1 \end{equation*}$$ Since coordinates are uniformly and independently selected, the $z^2$ component will make roughly a third of this. Hence: $$\begin{equation*} E[z^2] = \frac13 \end{equation*}$$

To find $E[z]$ we can imagine the unit sphere. The average of all coordinates will be in the exact center at $(0, 0, 0)$, giving us our expected value of z. Therefore:

$$\begin{equation*} Var(z) = \frac13 - 0^2 = \boxed{\frac13} \end{equation*}$$