{"schema_version":"ocme-site-results-v1","locale":"en","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-expectation"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-expectation-01","title":{"zh-Hant":"有限模型的期望值：線性性的證明與相依變數的實例","en":"Expectation in finite models: proving linearity, with an instance of dependent variables"},"summary":{"zh-Hant":"在非空有限樣本空間 S、非負且總和為 1 的權重 w_s 下，證明 E[c]=c、E[aX+b]=aE[X]+b、E[X+Y]=E[X]+E[Y]（X、Y 定義在同一個 S 上，不假設獨立）。再在 S={1,2,3}、w₁=1/2、w₂=1/4、w₃=1/4，X=(2,−2,0)、Y=(1,3,−1) 上計算 E[X]、E[Y]、E[X+Y]、E[−3X+5]，說明 X、Y 不獨立而線性性仍成立，並以 E[XY]≠E[X]E[Y] 說明線性性不推出乘積公式。","en":"For a nonempty finite sample space S with nonnegative weights w_s summing to 1, prove E[c]=c, E[aX+b]=aE[X]+b and E[X+Y]=E[X]+E[Y] (X, Y on the same S, independence not assumed). Then on S={1,2,3} with w₁=1/2, w₂=1/4, w₃=1/4, X=(2,−2,0), Y=(1,3,−1), compute E[X], E[Y], E[X+Y] and E[−3X+5], show that X and Y are not independent while linearity still holds, and use E[XY]≠E[X]E[Y] to show that linearity does not give a product formula."},"mko_ids":["mko-expectation"]}],"title":"Concept derivation guides"}