Probability and Statistics

combinatorics

<script type="math/tex; mode=display" id="MathJax-Element-1">C(N,K)=\frac{N!}{K!(N-K)!}</script>

Let C(N, K) = 1 for K = 0 or K = N, and C(N, K) = C(N − 1, K) + C(N − 1, K − 1) for N ≥ 1.
Prove that C(N, K) = N!/(K!(N−K)!) for N ≥ 1 and 0 ≤ K ≤ N.

counting

1. 掷10次硬币，4次正面朝上的概率是 C(10,4)∗(12)10 <script type="math/tex" id="MathJax-Element-2">C(10,4)*(\frac{1}{2})^{10}</script>
2. 从52张扑克里随机挑选5张，组成“XXXYY”的概率是 C(4,3)C(4,2)A(5,5)A(52,5) <script type="math/tex" id="MathJax-Element-3">\frac{C(4,3)C(4,2)A(5,5)}{A(52,5)}</script>

conditional probability

掷一枚硬币三次，已知其中一次正面朝上，求三次全部正面朝上的概率 0.531−0.53 <script type="math/tex" id="MathJax-Element-4">\frac{0.5^3}{1-0.5^3}</script>

bayes theorem

A program selects a random integer X like this: a random bit is first generated uniformly.
If the bit is 0, X is drawn uniformly from {0, 1, … , 7}; otherwise, X is drawn uniformly from {0, −1, −2, −3}.
If we get an X from the program with |X| = 1, what is the probability that X is negative?

union/intersection

If P(A) = 0.3 and P(B) = 0.4,
what is the maximum possible value of P(A ∩ B)?
0.3
what is the minimum possible value of P(A ∩ B)?
0
what is the maximum possible value of P(A ∪ B)?
0.7
what is the minimum possible value of P(A ∪ B)?
0.4

linear algebra

rank

2

inverse

[A|E] -> [E|A^-1]

[0.125 -0.625 0.750
-0.25 0.75 -0.50
0.375 -0.375 0.250 ]

eigenvalues/eigenvectors

singular value decomposition

PD/PSD

inner product

calculus

differential and partial differential

chain rule

gradient and Hessian

taylor’s expansion

optimization

vector calculus

参考资料

1. 机器学习基石HomeWork0