Discrete type ࿈ଓܕ Continuous type ֬ Probability ֬ม Random variable ֬ Probability distribution ֬ີؔ Probability density function: PDF ظ Expected value ظͷઢܗੑ Linearity of expectation ͖݅ͭ֬ Conditional probability ಉ࣌֬ Joint probability ඪ४Խ Standardization ಠཱੑ Independent ͖݅ͭಠཱ Conditional independent શ֬ͷެࣜ Law of total probability पล֬ Marginal probability ࿈ެࣜ Chain rule A Pr(X), Pr(X = x) X Pr f E[X] Pr[Y |A = a] Pr[Y |A], Pr(Y = y, A = a) Y⊥ ⊥ A|L Y⊥ ⊥ A
a Y = y k Outcome Y Sex A n Joint probability 1 1 (M) 20 20/200 = 0.1 1 0 (F) 50 50/200 = 0.25 0 1 (M) 80 80/200 = 0.4 0 0 (F) 50 50/200 = 0.25 Pr(Y, A) ಉ࣌֬ɿ ෳͷ֬ม͕ಉ࣌ʹى͜Δ֬ k Outcome Y Sex A = 1 n Conditional probability 1 1 20 20/100 = 0.2 0 1 80 80/100 = 0.8 Pr(Y |A = 1) k Outcome Y Sex A = 0 n Conditional probability 1 0 50 50/100 = 0.5 0 0 50 50/100 = 0.5 Pr(Y |A = 0) αϒάϧʔϓ
Y A L Pr(Y, A, L) = Pr(Y|L) Pr(L|A) Pr(A) A Y Pr(Y, A) = Pr(Y|A) Pr(A) A Y Pr(Y, A) = Pr(Y) Pr(A) Y A L Pr(Y, A, L) = Pr(Y|A, L) Pr(A|L) Pr(L) Pr(Y, A, L) = Pr(Y|A, L) Pr(L, A) Pr(A) = Pr(Y|A, L) Pr(L|A) Pr(A) L Y A Pr(Y, A, L) = Pr(L|Y, A) Pr(Y) Pr(A)