A vector with coordinates X-Y is X times I hat, plus Y times j hat.

坐标为X-Y的向量等于X乘以I帽 加上Y乘以j帽。

Mean of x, times mean of y, mean of x times mean of x.

x 的平均值乘以 y 的平均值，x 的平均值乘以 x 的平均值。

And you could even view it as this, over the mean of the X times the X's.

您甚至可以将其视为 X 乘以 X 的平均值。

Expected value of X times the expected value of Y. And let's see if we can simplify it right here.

X 的期望值乘以 Y 的期望值。让我们看看是否可以在这里简化它。

So the expected value of our random variable X times our random variable Y minus-- well, I'll just do the X first.

所以我们的随机变量 X 的期望值乘以我们的随机变量 Y 减去——好吧，我先做 X。

Well, if we take this whole sum and we divide it by n terms, this is going to be the mean value for x times y.

好吧，如果我们把这个总和除以 n 项， 这将是 x 乘以 y 的平均值。

This whole thing is about working out one plus X times itself a certain number of times, but Newton says, screw that just apply the theorem.

这整件事是关于计算出一加X乘以一定次数的自己，但是牛顿说，那就算了吧，只要应用定理就可以了。

So that's just going to be plus-- I'll freeze this-- expected value of X times the expected value of Y. Now what do we have here?

所以这就是加上——我将冻结这个——X 的预期值乘以 Y 的预期值。现在我们这里有什么？

That's equal to the limit as x approaches a of f of x, times the limit as x approaches a of g of x.

这等于当 x 接近 x 的 f 的 a 时的极限乘以 x 接近 x 的 g 的 a 时的极限。

If you multiply both the numerator and denominator by negative 1, you could see this written as the mean of the xy's minus the mean of x times the mean of the y's.

如果将分子和分母都乘以负数 1，您会看到这写成 xy 的平均值减去 x 的平均值乘以 y 的平均值。

So plus X times the negative expected value of Y. So I'll just say minus X times the expected value of Y. And that negative sign comes from this negative sign right over here.

所以加上 X 乘以 Y 的负预期值。所以我只说负 X 乘以 Y 的预期值。负号来自这里的负号。

We have expected value of Y times the expected value of X. And then we are subtracting the expected value of X times the expected value of Y. These two things are the exact same thing.

我们有 Y 的期望值乘以 X 的期望值。然后我们减去 X 的期望值乘以 Y 的期望值。这两个东西是完全一样的。

So if we multiply the numerator by the mean of the x's, we get the mean of the x's times the mean of the y's minus, this and this will cancel out, minus the mean of the xy's.

因此，如果我们将分子乘以 x 的平均值，我们得到 x 的平均值乘以 y 的平均值减去， 这个和这个将抵消， 减去 xy 的平均值。

If you change this Y to an X, this becomes X minus the expected value of X times X minus expected value of X. Or that's the expected value of X minus the expected value of X squared.

如果将此 Y 更改为 X，则变为 X 减去 X 的期望值乘以 X 减去 X 的期望值。 或者这就是 X 的期望值减去 X 的期望值的平方。

X and Y are equal to the expected value of-- I'll switch back to my colors just because this is the final result-- the expected value of X times the expected value of the product of XY minus-- what is this?

X 和 Y 等于预期值——我将切换回我的颜色只是因为这是最终结果——X 的预期值乘以 XY 乘积的预期值减去——这是什么？

And then we have minus expected value of X times Y, just doing the distributive property twice, and then finally you have the negative expected value of X times a negative expected value of Y. And the negatives cancel out.

然后我们得到 X 乘以 Y 的负期望值，只需执行两次分配属性，最后您得到 X 的负期望值乘以 Y 的负期望值。 负数抵消了。