Imperialism: a policy of extending a country’s power and influence though diplomacy or military force.

1. Colonies
2. Protectorate — nations has own government legally controlled by outside power
3. Sphere of influence

U.S. Imperialism, why?

1. “Desire for Military strength”: for a nation to be an international player, you have to have a strong navy
2. “Thirst for new markets”: if we continue to expand, we will have more economic power
3. “Belief in supernatural superiority”: trust that own culture is better

Alaska — “Seward’s Ice Box”, purchased from czarist Russia.

Key insight: suppose you have some fairly rare event and you want the likelihood of it. We can do this by drawing normal samples and reweighing them.

Suppose we want \(p_{\text{fail}}\); and we have \(q\) the proposal distribution and \(p\) the nominal distribution:

\(\tau \sim q\qty(\cdot)\), \(p_{\text{fail}} = \int 1 \qty {\tau \not\in \psi} p\qty(\tau) \dd{\tau }\)

What if we define a weird \(1\) such that:

\begin{equation} 1 = \frac{q\qty(\tau)}{q\qty(\tau)} \end{equation}

The Inbox is an Inbox for quick captures.

If an outcome can be from sets \(A=m\) or \(B=n\) with no overlaps, where \(A \cap B = \emptyset\), then, the total number of outcomes are \(|A| + |B| = m+n\)

If there are overlap:

\begin{equation} N = |A|+|B| - |A \cap B| \end{equation}

\(n\) random random variables are IID if they are

1. independent
2. identically distributed (see below)

“identically distributed”

Consider \(n\) random variables:

• \(X_i\) all have the same PMF / PDF
• and therefore, all have the same expectation and variance

central limit theorem

when things are IID, you can use central limit theorem.