AP Statistics is an examination by the CollegeBoard.

See also crap to remember for AP Stats

Non-Focus Mistakes

• file:///Users/houliu/Documents/School Work/The Bible/APStats/APStats5Steps.pdf
• file:///Users/houliu/Documents/School Work/The Bible/APStats/APStats5Steps.pdf
• file:///Users/houliu/Documents/School Work/The Bible/APStats/APStats5Steps.pdf
• Interpretation of regression outputs

Backlog

• Chi-square
• file:///Users/houliu/Documents/School Work/The Bible/APStats/APStats5Steps.pdf
• file:///Users/houliu/Documents/School Work/The Bible/APStats/APStats5Steps.pdf

Notes

• confidence interval
• hypothesis testing
• t-statistics
• chi-square
• data inference
• binomial distribution

Show that:

\begin{equation} \dv t e^{tA} = e^{tA}A \end{equation}

We can apply the result we shown in eigenvalue:

\begin{equation} \dv t \qty(e^{tA}) = \dv t \qty(I + \sum_{k=1}^{\infty} \frac{t^{k}}{k!}A^{k}) = \qty(\sum_{k=1}^{\infty }\frac{1}{k!}kt^{k-1}A^{k-1})A \end{equation}

We do this separation because \(k=0\) would’t make sense to raise \(A\) (\(k-1=-1\)) to as we are unsure about the invertability of \(A\). Obviously \(\frac{1}{k!}k = \frac{1}{(k-1)!}\). Therefore, we can shift our index back yet again:

\begin{equation} \qty(\sum_{k=1}^{\infty }\frac{1}{k!}kt^{k-1}A^{k-1})A = \qty(\sum_{j=0}^{\infty }\frac{1}{j!}t^{j}A^{j})A \end{equation}

Direct Sampling

Direct Sampling is an approximate inference method where we pull samples from the given joint probability distribution.

Example

Suppose we are interested in:

where we dare \(P(B^{1}|D^{1},C^{1})\).

Step 1: sort

We obtain a topological sort of this network:

\begin{equation} B, S, E, D, C \end{equation}

Step 2: sample from \(B,S\)

• We sample \(B\). We sampled that \(B=1\) today.
• We sample \(S\). We sampled that \(S=0\) today.

Step 3: sample from \(E\)

• We sample \(E\) GIVEN what we already sampled, that \(B=1, S=0\), we sampled that that \(E = 1\)

Step 4: sample from \(D, C\)

• We sample \(D\) given that \(E=1\) as we sampled.
• We sample \(C\) given that \(E=1\) as we sampled.

Repeat

Repeat steps 2-4

How do we deal with Markov Decision Process solution with continuous state space?

Let there be a value function parameterized on \(\theta\):

\begin{equation} U_{\theta}(s) \end{equation}

Let us find the value-function policy of this utility:

\begin{equation} \pi(s) = \arg\max_{a} \qty(R(s,a) + \gamma \sum_{s’}^{} T(s’|s,a) U_{\theta}(s’)) \end{equation}

We now create a finite sampling of our state space, which maybe infinitely large (for instance, continuous):

\begin{equation} S \in \mathcal{S} \end{equation}

where, \(S\) is a set of discrete states \(\{s_1, \dots, s_{m}\}\).