Preface
Probability distributions
- Uniform
- Poisson
- Discrete
- Parameter: $\lambda$
- $P(X = k) = \frac {e^{-\lambda}\lambda^k}{k!}$
- Normal
- t-dis???
- Chi-square 卡方分布
- Cauchy
Benford’s Law
$P(d) = log_{10}(d+1)-log_{10}(d)$, where d is the case that the first digit of the data is d
Ch 1
Sample space: $S$
event:
- subset of $S$
- measurable, (can assign a probability)
- subset is NOT event
Conditional Probability:
$P(A|B) = \frac{P(A\cap B)}{P(B)}$
Bayes theorem
$P(A) = \sum{P(A\cap B_i)}$
Now: We want to know: if A happened, what are the p of different B?
$P(B_j|A) = \frac{P(A\cap B_j)}{P(A)}$
$P(A\cap B_j) = P(A|B_j)P(B_j)$
Bayes theorem definition:
Consider a partition $B_j$ of $S$ and event $A$:
$P(B_j|A) = \frac{P(A\cap B_j)}{P(A)}= \frac{P(A|B_j)P(B_j)}{\sum{P(A|B_i)P(B_i)}}$
Monty Hall Problem
$B_j = C_2|X_1, A = H_3|X_1$
Ch 2 Discrete Random Variables
Mean is the quantity a that minimizes $min_aE(X-a)^2$
Variance $(x-E(x))^2$
Moment generating function = mgf = summary of the overall random behavior
median $min_b |x-b|$