The document discusses several mathematical formulas and concepts. It introduces formulas for probability distributions, kernel density estimation, distance estimation, and kinematic equations. It also discusses properties of different estimators like consistency and discusses properties of angles and how they relate to each other.

2. 
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4. 𝑝 𝑛−1,𝑛 = Pr(𝑂𝑛 = 𝑜 𝑛|𝑂 𝑛−1 = 𝑜 𝑛−1, … , 𝑂1 = 𝑜1),
𝑝 𝑛−1,𝑛 (𝑛 − 1) 𝑛
𝑂𝑛 𝑛 𝑜 𝑛

6. 𝑓̂𝑖,𝑗
ℎ
(𝑥) =
1
𝑛ℎ
∑ 𝛷(
𝑥 − 𝑑𝑖,𝑗,𝑘
ℎ
)
𝑛
𝑖=1
,
𝑓̂𝑖,𝑗
ℎ
(𝑥) 𝛷
𝑛 ℎ
max
𝒙,𝒚
∑ ln 𝑓̂𝑖,𝑗
ℎ
(√(𝑥𝑖 − 𝑥𝑗)2 + (𝑦𝑖 − 𝑦𝑗)2)
𝑖,𝑗
.
𝑥𝑖, 𝑦𝑖 𝑖

7. 𝑑̂ 𝐴,𝐵 = 𝑑 𝐴,𝐵 + 𝑓𝐴(𝑑̂ 𝐴,𝐵) + 𝑓𝐵(𝑑̂ 𝐴,𝐵),
𝑑 𝐴,𝐵 𝐴 𝐵 𝑓𝐴 𝑓𝐵 𝐴 𝐵
𝑑̂ 𝐴,𝐵
𝑑 𝐴,𝐵 = 𝑑̂ 𝐴,𝐵 + 𝑓𝐴,𝐵(𝑑̂ 𝐴,𝐵)
𝑓𝐴,𝐵 𝐴 𝐵 𝑓𝐴, 𝑓𝐵 𝑓𝐴,𝐵

8. 𝑲 𝒄
𝑲𝒍
𝑲 𝒑
𝑲 𝒄 𝑻
𝑲 𝒄
𝑛 ≥ 3 𝑚 ≥ 𝑛 𝑲𝒍 𝑻
𝑲𝒍
𝑛 ≥ 5 𝑚 ≥ 2𝑛 𝑲 𝒑 𝑻
𝑲 𝒑
𝑛 ≥ 7 𝑚 ≥ 3𝑛 𝑛 𝑚
∎

12. ∆𝑥 = 𝑣 𝑥 ∗ ∆𝑡 = cos 𝛼 ∗ 𝑣 ∗ ∆𝑡,
∆𝑦 = 𝑣 𝑦 ∗ ∆𝑡 = sin 𝛼 ∗ 𝑣 ∗ ∆𝑡,
∆𝑥, ∆𝑦 𝑣 𝑥, 𝑣 𝑦
∆𝑡
𝛼

13. 𝛼
𝛼
𝛼
𝛼

15. α α α
α α α
α α α