Welcome!

class: center, middle, inverse, title-slide

.title[
# Welcome!
]
.author[
### Yue Jiang
]
.date[
### STA 210 / Duke University / Spring 2024
]

---

### Tracking Ceres

<img src="img/ceres.jpg" width="60%" style="display: block; margin: auto;" />

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### Carl Friedrich Gauss

<img src="img/deutschemark.png" width="100%" style="display: block; margin: auto;" />

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### *Theoria motus corporum coelestium*

<img src="img/gauss2.png" width="60%" style="display: block; margin: auto;" />

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### *Theoria motus corporum coelestium*

<img src="img/gaussa.png" width="90%" style="display: block; margin: auto;" />

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### *Theoria motus corporum coelestium*

<img src="img/gaussb.png" width="90%" style="display: block; margin: auto;" />

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### The problem at hand

<img src="img/gauss1.png" width="60%" style="display: block; margin: auto;" /><img src="img/gauss3.png" width="60%" style="display: block; margin: auto;" />

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### The solution

<img src="img/gaussc.png" width="90%" style="display: block; margin: auto;" />

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### Prediction vs. Explanation

<img src="img/gauss1.png" width="60%" style="display: block; margin: auto;" /><img src="img/gauss3.png" width="60%" style="display: block; margin: auto;" />

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### Prediction vs. Explanation

<img src="img/fire.png" width="90%" style="display: block; margin: auto;" />

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### Association vs. Causation

<img src="img/fire.png" width="90%" style="display: block; margin: auto;" />

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### Data-driven vs. Theory-driven

<img src="img/experiment.png" width="90%" style="display: block; margin: auto;" />

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### Prospective vs. Retrospective

<img src="img/ikea.png" width="90%" style="display: block; margin: auto;" />

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### Why should we care about linear models?

<img src="img/eniac.png" width="90%" style="display: block; margin: auto;" />

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### Linear models are foundational

<img src="img/trading.png" width="90%" style="display: block; margin: auto;" />

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### Linear models are flexible

<img src="img/cholera.png" width="90%" style="display: block; margin: auto;" />

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### Linear models are transparent

<img src="img/scatter2.png" width="90%" style="display: block; margin: auto;" />

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### Linear models are translational

<img src="img/mnist.png" width="90%" style="display: block; margin: auto;" />

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### Linear models are useful

<img src="img/basketball.png" width="90%" style="display: block; margin: auto;" />

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### How does Y change as X varies?

<img src="img/scatter.png" width="90%" style="display: block; margin: auto;" />

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### Means and variances

<img src="img/hurricane.png" width="90%" style="display: block; margin: auto;" />

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### Means and variances

.vocab[Expectations] are *"averages"* :

`\begin{align*}
E(Y) &= \sum_{y} y \times p_Y(y)
\end{align*}`

.vocab[Variances] are *"expected squared deviations around the mean"* :

`\begin{align*}
Var(Y) &= E\left[ \left(Y - E(Y)\right)^2 \right]
\end{align*}`

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### Means and variances

.vocab[Conditional] means and variances, for instance

`\begin{align*}
E(Y | X = x) = \textrm{...some function of x} 
\end{align*}`

`\begin{align*}
Var(Y | X = x) = \textrm{...some other function of x}
\end{align*}`

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### How does Y change as X varies?

Well, how does the .vocab[distribution] of Y change as X varies?

<img src="img/scatter.png" width="45%" style="display: block; margin: auto;" />

Maybe...

`\begin{align*}
E(Y | X = x) = \beta_0 + \beta_1x
\end{align*}`

`\begin{align*}
Var(Y | X = x) = \sigma^2
\end{align*}`

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### Looking ahead

<img src="img/meteora.jpg" width="90%" style="display: block; margin: auto;" />

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### Some housekeeping

<img src="img/website.png" width="90%" style="display: block; margin: auto;" />