ML基石_HW0

• Probability and Statistics
  • combinatorics
  • counting
  • conditional probability
  • bayes theorem
  • unionintersection
• linear algebra
  • rank
  • inverse
  • eigenvalueseigenvectors
  • singular value decomposition
  • PDPSD
  • inner product
• calculus
  • differential and partial differential
  • chain rule
  • gradient and Hessian
  • taylors expansion
  • optimization
  • vector calculus
• 参考资料

Probability and Statistics

combinatorics

C(N,K)=N!K!(N−K)!

C(N,K)=\frac{N!}{K!(N-K)!}

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)10C(10,4)*(\frac{1}{2})^{10}
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)\frac{C(4,3)C(4,2)A(5,5)}{A(52,5)}

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\frac{0.5^3}{1-0.5^3}

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