线性回归分析-HW1

作业题目

1.

The regression effect: the brother of a high man is expected not to be as high as his brother, but a little lower; while the brother of a short man is expected not to be as short as his brother, but a little higher. There is a tendency toward the mean height.

2.

A

OM: $y=\frac{MX}{OX}x$

ON: $y=\frac{OY}{NY}x$

B

AB 对应 OL

CD 对应 OM

3.

A

mid-parent population 对应 1

指的是 1 和 5 同分布

B

经过斜槽后仍为正态分布。

子代整体为什么是正态分布：可以把子代看成相对母代多了几层的高尔顿板，因为高尔顿板最终近似呈现正态分布，所以子代整体也是正态分布。

是否是小正态分布组成大正态分布：是。可以把母代的每个槽看作是高尔顿板的一个起点，从每一个起点下落的小球，其最终位置分布都近似正态分布，而子代整体的分布就是每个起点对应正态分布加总的结果。

4

Mathematically, let

$$\begin{align} X & = \text{the position of a ball in the middle compartment} \\ Y & = \text{the position of a ball in the lower compartment} \end{align}$$

and suppose $X\sim N(0,\sigma_{1}^{2})$，$Y\sim N(0,\sigma_{2}^{2})$, where $\sigma_{1}<\sigma_2$. Let $\Delta=Y-X$, as discussed in class, we have $\Delta\sim N(0,\sigma^{2})$ and $\Delta \perp X$.

For the left panel, mathematically it is saying

$$\mathbb{E}[Y\mid X=x]=x.$$

For the right panel, it is saying

$$\mathbb{E}[X\mid Y=y]=cy, \text{where } 0<c<1$$

In fact, $c=\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}$. See 多元正态分布 › 条件期望

Intuitively, given the position in the lower compartment, though balls right above are more likely to reach this position, but the number of them is smaller than those in the middle. So on average, the original position is closer to the middle.

5

Let $X\sim N(65,1), Y\sim N(68,4), X\perp Y$, then $X-Y\sim N(-3,5)$

$$\begin{align} P(X>Y) & = P(X-Y>0) \\ & = 1-\Phi\left( \frac{3}{\sqrt{ 5 }} \right) \\ & \approx 0.09 \end{align}$$