找出方差

$Var(X)$ 的公式为 $E(X^2) - E(X)^2$

为了找到 $E(X^2)$，我们可以取所有 $f(x) \cdot X^2$ 的总和。在对连续分布上的数量求和时，积分是一个有用的工具。

$E(X^2)$ = $\int_0^1f(x)X^2 dx= \int_0^1 1\cdot X^2 dx = \frac13X^3 \Big|_0^1 = \frac13$

我们可以用类似的方式计算 $E(X)^2$，先对 $f(x)X$ 求和，然后对结果求平方，但更简单的方法是认识到均匀分布的平均结果将只是其中点。因此，$E(X)^2 = (\frac12)^2 = \frac14$

$Var(X) = \frac13 - \frac14 = \frac1{12}$

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Original Explanation

The formula for $Var(X)$ is $E(X^2) - E(X)^2$

To find $E(X^2)$, we can take the sum of all $f(x) \cdot X^2$. The integral is a useful tool to use when summing quantities over a continuous distribution.

$E(X^2)$ = $\int_0^1f(x)X^2 dx= \int_0^1 1\cdot X^2 dx = \frac13X^3 \Big|_0^1 = \frac13$

We can compute $E(X)^2$ in a similar way buy summing $f(x)X$ and then squaring that result, but a simpler way would be to realize that the average result of a uniform distribution will just be its midpoint. Hence, $E(X)^2 = (\frac12)^2 = \frac14$

$Var(X) = \frac13 - \frac14 = \frac1{12}$