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For each $i$ compute $f\left(\Psi\left(t_{i}\right)\right) \delta t_{i}$. A little algebraic trickery yields
\begin{aligned} f\left(\Psi\left(t_{i}\right)\right) \delta t_{i} &=f\left(\Psi\left(t_{i}\right)\right) \delta t_{i} \frac{\Delta t_{i}}{\Delta t_{i}} \ &=f\left(\Psi\left(t_{i}\right)\right) \frac{\delta t_{i}}{\Delta t_{i}} \Delta t_{i} \end{aligned}
Sum over all $i$.
The integral $\int_{C} f(x, y) d \mathbf{s}$ is defined to be the limit of the sum from the previous step as $n \rightarrow \infty$. But as $n \rightarrow \infty$ it also follows that $\Delta t_{i} \rightarrow 0$. Our integrand contains the term $\frac{\delta_{i}}{\Delta t_{i}}$. As $\Delta t_{i} \rightarrow \infty$ this converges to $\left|\frac{\partial \Psi}{\partial t}\right|$. Hence, our integral has become
$$\int_{C} f(x, y) d \mathbf{s}=\int_{a}^{b} f(\Psi(t))\left|\frac{\partial \Psi}{\partial t}\right| d t$$

MAST90123/STAT 550/Math 776 COURSE NOTES ：

We now compute the area of each parallelogram.
Observe that each parallelogram is spanned by the vectors
$$V_{u}=\Psi\left(u_{i+1}, v_{j}\right)-\Psi\left(u_{i}, v_{j}\right)$$
and
$$V_{v}=\Psi\left(u_{i}, v_{j+1}\right)-\Psi\left(u_{i}, v_{j}\right)$$
The desired area is thus the magnitude of the cross product of these vectors:
$$\text { Area }=\left|V_{u} \times V_{v}\right|$$
We now do some algebraic tricks:
\begin{aligned} \left|V_{u} \times V_{v}\right| &=\left|V_{u} \times V_{v}\right| \frac{\Delta u \Delta v}{\Delta u \Delta v} \ &=\left|\frac{V_{u}}{\Delta u} \times \frac{V_{v}}{\Delta v}\right| \Delta u \Delta v \end{aligned}

高等数理统计学 | Advanced Mathematical Statistics代写 STAT3056代考

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$$\pi_{i j}=\pi_{i} \pi_{j}$$
or
$$\log \pi_{i j}=\log \pi_{i}+\log \pi_{j}$$

$$\log \pi_{i j}=\alpha_{i}+\beta_{j}+\gamma_{i j}$$
This form mimics the additive analysis of variance models introduced in chapter 12. The idea can readily be extended to higher-order tables. For example, a possible model for a three-way table is
$$\log \pi_{i j k}=\alpha_{i}+\beta_{j}+\delta_{i j}+\epsilon_{i k}+\gamma_{j k}$$

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STAT3056COURSE NOTES ：

$$v_{i}=\frac{y_{i}-\bar{y}}{\sqrt{s_{y y}}}$$
We then have $s_{u u}=s_{v v}=1$ and $s_{u v}=r$. The least squares line for predicting $v$ from $u$ thus has slope $r$ and intercept
$$\bar{\beta}{0}=\bar{v}-r \bar{u}=0$$ and the predicted values are $$\hat{v}{i}=r u_{i}$$