{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-random-variable-independence"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-random-variable-independence-01","title":{"zh-Hant":"隨機變數獨立：完整核對有限聯合表，零協方差只是後果","en":"Independence of random variables: checking the complete finite joint table, with zero covariance only a consequence"},"summary":{"zh-Hant":"（一）聯合表 J₁：X∈{0,1}、Y∈{0,1,2}，P(0,0)=1/12、P(0,1)=1/12、P(0,2)=1/6、P(1,0)=1/6、P(1,1)=1/6、P(1,2)=1/3。由聯合表聚合兩個邊際，核對全部 6 個值對的乘積式，判定 X、Y 是否獨立；並證明獨立推出任意座標事件 {X∈S} 與 {Y∈T} 獨立與 E[XY]=E[X]E[Y]。（二）聯合表 J₂：X 在 {−1,0,1} 上均勻、Y=X²。證明 Cov(X,Y)=0 但 X、Y 不獨立。（三）聯合表 J₃ 說明只核一個值對不夠。","en":"(1) Joint table J₁: X∈{0,1}, Y∈{0,1,2}, P(0,0)=1/12, P(0,1)=1/12, P(0,2)=1/6, P(1,0)=1/6, P(1,1)=1/6, P(1,2)=1/3. Aggregate both marginals from the joint table, check the product equation at all 6 value pairs, and decide whether X and Y are independent; also prove that independence implies independence of any coordinate events {X∈S} and {Y∈T} and E[XY]=E[X]E[Y]. (2) Joint table J₂: X uniform on {−1,0,1} and Y=X². Prove Cov(X,Y)=0 although X and Y are not independent. (3) Use joint table J₃ to show that checking a single value pair is not enough."},"mko_ids":["mko-random-variable-independence"]}],"title":"概念推導導讀"}