\uc774 \ud3ec\uc2a4\ud305\uc5d0\uc11c\ub294 \uce21\ub3c4\ub860\uc744 \uae30\ubc18\uc73c\ub85c \ud655\ub960\uacfc \uad00\ub828\ub41c \uac1c\ub150\uc744 \uc815\uc758\ud558\uace0 \ud655\ub960\ubcc0\uc218\uc758 \ub3c5\ub9bd\uc131\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n
\\(\\varOmega\\)\uac00 \uc9d1\ud569\uc774\uace0 \\(\\mathcal{F}\\)\uac00 \\(\\varOmega\\)\uc758 \ubd80\ubd84\uc9d1\ud569\ub4e4\uc758 \\(\\sigma\\)-\ub300\uc218\uc774\uba70 \\(P\\)\uac00 \\(\\mathcal{F}\\) \uc704\uc5d0\uc11c\uc758 \uce21\ub3c4\uc774\uace0 \\(P(\\varOmega ) = 1\\)\uc77c \ub54c, \\((\\varOmega ,\\, \\mathcal{F} ,\\, P)\\)\ub97c \ud655\ub960\uacf5\uac04<\/span>(probability space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc5ec\uae30\uc11c \\(P\\)\ub97c \ud655\ub960\uce21\ub3c4<\/span>(probability measure) \ub610\ub294 \uac04\ub2e8\ud788 \ud655\ub960<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70, \\(\\mathcal{F}\\)\uc758 \uc6d0\uc18c\ub97c \uc0ac\uac74<\/span>(event)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n \\(B\\)\uac00 \uc0ac\uac74\uc774\uace0 \\(P(B) > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub450 \uc0ac\uac74 \\(A\\)\uc640 \\(B\\)\uc5d0 \ub300\ud558\uc5ec \\(\\mathcal{F}_1 ,\\) \\(\\cdots ,\\) \\(\\mathcal{F}_n\\)\uc774 \ud655\ub960\uacf5\uac04 \\((\\varOmega ,\\,\\mathcal{F} ,\\,P)\\) \uc704\uc5d0 \uc815\uc758\ub41c \\(\\sigma\\)-\ub300\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub4e4 \\(\\sigma\\)-\ub300\uc218\ub4e4\uc774 \ub3c5\ub9bd<\/span>\uc774\ub77c\ub294 \uac83\uc740 \\(\\left\\{ 1,\\,2,\\,\\cdots,\\,n\\right\\}\\)\uc5d0\uc11c \uc120\ud0dd\ud55c \uc11c\ub85c \ub2e4\ub978 \\(k\\)\uac1c\uc758 \uc784\uc758\uc758 \ucca8\uc790 \\(i_1 ,\\) \\(i_2 ,\\) \\(\\cdots ,\\) \\(i_k\\)\uc640 \uc784\uc758\uc758 \ubd80\ubd84\uc9d1\ud569 \\(F_{i_n} \\subseteq \\mathcal{F}_{i_n}\\)\uc5d0 \ub300\ud558\uc5ec \\((\\varOmega ,\\,\\mathcal{F} ,\\,P)\\)\uac00 \ud655\ub960\uacf5\uac04\uc774\uace0 \\(X : \\varOmega \\,\\to\\,\\mathbb{R}\\)\uac00 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(a\\in\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec \\(X^{-1} ([a,\\,\\infty )) \\in \\mathcal{F}\\)\uc774\uba74 \\(X\\)\ub97c \ud655\ub960\ubcc0\uc218<\/span>(random variable)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\varOmega \\subseteq \\mathbb{R}\\)\uac00 \uac00\uce21\uc9d1\ud569\uc774\uace0 \\(\\mathcal{F} = \\mathcal{B}\\)\uac00 \\(\\varOmega\\)\uc758 \ubcf4\ub810 \ubd80\ubd84\uc9d1\ud569\uc758 \\(\\sigma\\)-\ub300\uc218\uc774\uba74 \ud655\ub960\ubcc0\uc218\ub294 \\(\\mathbb{R}\\)\ub85c\ubd80\ud130 \\(\\mathbb{R}\\)\ub85c\uc758 \ubcf4\ub810 \ud568\uc218\uc774\ub2e4.<\/p>\n \ud655\ub960\ubcc0\uc218\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(X\\)\uac00 \uc784\uc758\uc758 \ud655\ub960\ubcc0\uc218\uc774\uace0 \\(B\\)\uac00 \ubcf4\ub810 \uc9d1\ud569\uc77c \ub54c \uc0ac\uac74\uc758 \ub3c5\ub9bd\uc131\uc744 \uc815\uc758\ud55c \uac83\ucc98\ub7fc \ud655\ub960\ubcc0\uc218\uc758 \ub3c5\ub9bd\uc131\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \\(X\\)\uc640 \\(Y\\)\uac00 \ud655\ub960\ubcc0\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(X,\\) \\(Y\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \\(\\sigma\\)-\ub300\uc218\uac00 \uc11c\ub85c \ub3c5\ub9bd\uc774\uba74 \u2018\\(X\\)\uc640 \\(Y\\)\ub294 \ub3c5\ub9bd<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc989 \\(X\\)\uc640 \\(Y\\)\uac00 \ub3c5\ub9bd\uc774\ub77c\ub294 \uac83\uc740 \\(\\mathbb{R}\\)\uc758 \uc784\uc758\uc758 \ubcf4\ub810 \uc9d1\ud569 \\(B,\\) \\(C\\)\uc5d0 \ub300\ud558\uc5ec \ub3c5\ub9bd\uc774 \uc544\ub2cc \ub450 \uc0ac\uac74\uc744 \uc885\uc18d\uc0ac\uac74<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \ub3c5\ub9bd\uc774 \uc544\ub2cc \ub450 \ud655\ub960\ubcc0\uc218\ub97c \uc885\uc18d \ud655\ub960\ubcc0\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n \\(X\\)\uac00 \ud655\ub960\uacf5\uac04 \\((\\varOmega ,\\,\\mathcal{F} ,\\,P )\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ud655\ub960\ubcc0\uc218\uc77c \ub54c \\(X\\)\uc758 \uae30\ub313\uac12<\/span>(expectation)\uc744 \ud655\ub960\ubcc0\uc218 \\(X\\)\uc640 \\(t\\in \\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec \\(n\\)\uc774 \uc790\uc5f0\uc218\ub77c\uace0 \ud558\uc790. \ud655\ub960\ubcc0\uc218 \\(X\\in L^n (\\varOmega )\\)\uc758 \\(n\\)\ucc28 \ubaa8\uba58\ud2b8<\/span>(moment of order n)\ub780 \\(\\mathbb{E} (X^n )\\)\uc744 \uc758\ubbf8\ud55c\ub2e4. \ud2b9\ud788 1\ucc28 \ubaa8\uba58\ud2b8\ub97c \uae30\ub313\uac12\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\mathbb{E} (X) = \\mu\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(X\\)\uc758 \uc911\uc2ec\ubaa8\uba58\ud2b8<\/span>(central moment)\ub780 \\(\\mathbb{E} (X-\\mu )^n\\)\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n \ubaa8\uba58\ud2b8\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ud655\ub960\ubd84\ud3ec\ub97c \uc774\uc6a9\ud558\uc5ec \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \ub450 \ud655\ub960\ubcc0\uc218\uac00 \ub3c5\ub9bd\uc774\uae30 \uc704\ud55c \uc870\uac74\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n \ud655\ub960\ubcc0\uc218\uac00 \ub3c5\ub9bd\uc774\uae30 \uc704\ud55c \uc870\uac74.<\/span><\/p>\n \ub450 \ud655\ub960\ubcc0\uc218 \\(X,\\) \\(Y\\)\uac00 \ub3c5\ub9bd\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubcf4\ub810 \uac00\uce21\uc778 \uc784\uc758\uc758 \uc720\uacc4 \ud568\uc218 \\(f,\\) \\(g\\)\uc5d0 \ub300\ud558\uc5ec \uc99d\uba85<\/p>\n (1)\uc774 \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(B_1 ,\\) \\(B_2\\)\uac00 \ubcf4\ub810 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(f = \\mathbf{1}_{B_1} ,\\) \\(g = \\mathbf{1}_{B_2}\\)\ub77c\uace0 \ud558\uace0 (1)\uc744 \uc774\uc6a9\ud558\uba74 \uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(X\\)\uc640 \\(Y\\)\uac00 \ub3c5\ub9bd\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(f = \\mathbf{1}_{B_1} ,\\) \\(g = \\mathbf{1}_{B_2}\\)\uc640 \ubcf4\ub810 \uc9d1\ud569 \\(B_1 ,\\) \\(B_2\\)\uc5d0 \ub300\ud558\uc5ec (1)\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub450 \ub2e8\uc21c\ud568\uc218 \uc774 \ud3ec\uc2a4\ud305\uc5d0\uc11c\ub294 \uce21\ub3c4\ub860\uc744 \uae30\ubc18\uc73c\ub85c \ud655\ub960\uacfc \uad00\ub828\ub41c \uac1c\ub150\uc744 \uc815\uc758\ud558\uace0 \ud655\ub960\ubcc0\uc218\uc758 \ub3c5\ub9bd\uc131\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ud655\ub960\uacf5\uac04\uacfc \ud655\ub960\uce21\ub3c4 \\(\\varOmega\\)\uac00 \uc9d1\ud569\uc774\uace0 \\(\\mathcal{F}\\)\uac00 \\(\\varOmega\\)\uc758 \ubd80\ubd84\uc9d1\ud569\ub4e4\uc758 \\(\\sigma\\)-\ub300\uc218\uc774\uba70 \\(P\\)\uac00 \\(\\mathcal{F}\\) \uc704\uc5d0\uc11c\uc758 \uce21\ub3c4\uc774\uace0 \\(P(\\varOmega ) = 1\\)\uc77c \ub54c, \\((\\varOmega ,\\, \\mathcal{F} ,\\, P)\\)\ub97c \ud655\ub960\uacf5\uac04(probability space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc5ec\uae30\uc11c \\(P\\)\ub97c \ud655\ub960\uce21\ub3c4(probability measure) \ub610\ub294 \uac04\ub2e8\ud788 \ud655\ub960\uc774\ub77c\uace0 \ubd80\ub974\uba70, \\(\\mathcal{F}\\)\uc758 \uc6d0\uc18c\ub97c \uc0ac\uac74(event)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(B\\)\uac00 \uc0ac\uac74\uc774\uace0 \\(P(B) > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\( P(A|B) := \\frac{P(A\\cap B)}{P(B)}\\) \ub97c \u2018\\(B\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[67],"tags":[],"class_list":["post-2478","post","type-post","status-publish","format-standard","hentry","category-probability-n-statistics"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2478","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=2478"}],"version-history":[{"count":16,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2478\/revisions"}],"predecessor-version":[{"id":6138,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2478\/revisions\/6138"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2478"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2478"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2478"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\\[ P(A|B) := \\frac{P(A\\cap B)}{P(B)}\\]
\n\ub97c \u2018\\(B\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \\(A\\)\uc758 \uc870\uac74\ubd80\ud655\ub960<\/span>(conditional probability of A given B)\u2019\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n\ub3c5\ub9bd\uacfc \uc885\uc18d\uc758 \uc815\uc758<\/h3>\n
\n\\[P(A \\cap B) = P(A) \\cdot P(B)\\]
\n\uac00 \uc131\ub9bd\ud560 \ub54c, \u2018\\(A\\)\uc640 \\(B\\)\ub294 \uc11c\ub85c \ub3c5\ub9bd<\/span>(independent)\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\uac83\uc740 \uc5ec\ub7ec \uac1c\uc758 \uc0ac\uac74\uc758 \ub3c5\ub9bd\uc131\uc73c\ub85c \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4. \uc0ac\uac74 \\(A_1 ,\\) \\(\\cdots ,\\) \\(A_n\\)\uc774 \ub3c5\ub9bd<\/span>\uc774\ub77c\ub294 \uac83\uc740 \\(k\\le n\\)\uc778 \uc784\uc758\uc758 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(k\\)\uac1c\uc758 \uc0ac\uac74\ub4e4\uc758 \uad50\uc9d1\ud569\uc758 \ud655\ub960\uc774 \uadf8 \uc0ac\uac74\ub4e4\uc758 \ud655\ub960\uc758 \uacf1\uacfc \uac19\uc740 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n
\n\\[P(F_{i_1} \\cap F_{i_2} \\cap \\cdots \\cap F_{i_k} ) = P(F_{i_1}) \\cdot P(F_{i_2}) \\cdots P(F_{i_k})\\]
\n\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n
\n\\[X^{-1}(\\mathcal{B}) = \\left\\{ S \\subseteq \\mathcal{F} \\,\\vert\\, S=X^{-1} (B) \\text{ for some } B\\in\\mathcal{B} \\right\\}\\]
\n\ub294 \\(\\mathcal{F}\\)\uc5d0 \ud3ec\ud568\ub418\ub294 \\(\\sigma\\)-\ub300\uc218\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \ud655\ub960\ubcc0\uc218 \\(X\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \\(\\sigma\\)-\ub300\uc218<\/span>\ub97c \\(\\mathcal{F}_X\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n
\n\\[P_X (B) := P(X^{-1} (B))\\]
\n\ub77c\uace0 \uc815\uc758\ud568\uc73c\ub85c\uc368 \\(B\\)\uc758 \\(\\sigma\\)-\ub300\uc218 \uc704\uc5d0\uc11c \uc815\uc758\ub41c \uce21\ub3c4\ub97c \uc5bb\ub294\ub2e4. \uc774 \uce21\ub3c4 \\(P_X\\)\ub97c \ud655\ub960\ubcc0\uc218 \\(X\\)\uc758 \ud655\ub960\ubd84\ud3ec<\/span>(probability distribution)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(P_X\\)\ub294 \uac00\uc0b0\uac00\ubc95\uc801\uc774\ub2e4. \ub354\uc6b1\uc774 \\(\\varOmega = \\mathbb{R} ,\\) \\(\\mathcal{F} = \\mathcal{B}\\)\uc77c \ub54c \\((\\mathbb{R} ,\\,\\mathcal{B} ,\\,P_X )\\)\ub294 \ud655\ub960\uacf5\uac04\uc774 \ub41c\ub2e4.<\/p>\n
\n\\[P(X^{-1} (B) \\cap Y^{-1} (C)) = P(X^{-1} (B)) P(Y^{-1} (C))\\]
\n\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n\uae30\ub313\uac12\uacfc \ubd84\uc0b0<\/h3>\n
\n\\[\\mathbb{E} (X) := \\int_{\\varOmega} X \\,dP\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4. \uae30\ub313\uac12\uc740 \ud655\ub960\ubd84\ud3ec\ub97c \uc774\uc6a9\ud558\uc5ec
\n\\[\\mathbb{E}(X) = \\int_{-\\infty}^{\\infty} x\\, dP_X (x)\\]
\n\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\uc73c\uba70, \uc808\ub300\uc5f0\uc18d\uc778 \ud655\ub960\ubcc0\uc218 \\(X\\)\uc758 \ubc00\ub3c4 \\(f_X\\)\ub97c \uc774\uc6a9\ud558\uc5ec
\n\\[\\mathbb{E}(X) = \\int_{-\\infty}^{\\infty} x f_X (x) \\,dx\\]
\n\ub85c \uacc4\uc0b0\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n
\n\\[\\varphi_X (t) = \\mathbb{E} (e^{i t X} )\\]
\n\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(\\varphi _X\\)\ub97c \\(X\\)\uc758 \ud2b9\uc131\ud568\uc218<\/span>(characteristic function of X)\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\uc131\ud568\uc218\ub294 \ud655\ub960\ubd84\ud3ec\ub97c \uc774\uc6a9\ud558\uc5ec
\n\\[\\varphi_X (t) = \\int e^{i t x} \\,dP_X (x)\\]
\n\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\uc73c\uba70, \uc808\ub300\uc5f0\uc18d\uc778 \ud655\ub960\ubcc0\uc218 \\(X\\)\uc758 \ubc00\ub3c4 \\(f_X\\)\ub97c \uc774\uc6a9\ud558\uc5ec
\n\\[\\varphi_X (t) = \\int e^{itx} f_X (x) \\,dx\\]
\n\ub85c \uacc4\uc0b0\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n
\n\\[\\begin{align}
\n\\mathbb{E} (X^n) &= \\int x^n \\,d P_X (x) ,\\\\[6pt]
\n\\mathbb{E} ((X-\\mu )^n ) &= \\int (x-\\mu )^n \\,d P_X (x) .
\n\\end{align}\\]
\n\ub9cc\uc57d \\(X\\)\uac00 \ubc00\ub3c4 \\(f_X\\)\ub97c \uac00\uc9c4\ub2e4\uba74 \ubaa8\uba58\ud2b8\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.
\n\\[\\begin{align}
\n\\mathbb{E} (X^n) &= \\int x^n f_X (x)\\,dx ,\\\\[6pt]
\n\\mathbb{E} ((X-\\mu )^n ) &= \\int (x-\\mu )^n f_X (x) \\,dx .
\n\\end{align}\\]
\n\ud655\ub960\ubcc0\uc218 \\(X\\)\uc758 2\ucc28 \ubaa8\uba58\ud2b8
\n\\[\\operatorname{Var} (X) = \\mathbb{E} (X-\\mathbb{E} (X))^2\\]
\n\uc744 \\(X\\)\uc758 \ubd84\uc0b0<\/span>(variance)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(X\\)\uc758 \ubd84\uc0b0\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.
\n\\[\\operatorname{Var} (X) = \\mathbb{E}(X^2 ) – (\\mathbb{E} (X))^2 .\\]
\n\uac00\uc6b0\uc2a4 \ubd84\ud3ec\uc758 \uae30\ub313\uac12\uacfc \ubd84\uc0b0\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.
\n\\[\\begin{align}
\n\\mu &= \\frac{1}{\\sqrt{2\\pi}\\sigma} \\int_{\\mathbb{R}} xe^{- \\frac{(x-\\mu )^2}{2\\sigma ^2}} dx , \\\\[6pt]
\n\\sigma^2 &= \\frac{1}{\\sqrt{2\\pi}\\sigma} \\int_{\\mathbb{R}} (x-\\mu )^2 e^{- \\frac{(x-\\mu )^2}{2\\sigma ^2}} dx.
\n\\end{align}\\]<\/p>\n\ub3c5\ub9bd\uc131\uc758 \uc870\uac74<\/h3>\n
\n\\[\\mathbb{E}(f(X)g(Y)) = \\mathbb{E}(f(X))\\mathbb{E}(g(Y)) \\tag{1}\\]
\n\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n
\n\\[\\begin{align}
\n\\int_{\\varOmega} & \\mathbf{1}_{B_1} (X(\\omega ))\\mathbf{1}_{B_2} (Y(\\omega ))\\,d P(\\omega ) \\\\[6pt]
\n&= \\int_{\\varOmega} \\mathbf{1}_{B_1} (X(\\omega )) \\,dP(\\omega ) \\int_{\\varOmega} \\mathbf{1}_{B_2} (Y(\\omega )) \\,dP(\\omega )
\n\\end{align}\\]
\n\ub97c \uc5bb\ub294\ub2e4. \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc744 \uacc4\uc0b0\ud558\uba74
\n\\[\\begin{align}
\n\\int_{\\varOmega} & \\mathbf{1}_{B_1 \\times B_2} (X(\\omega) ,\\, (Y(\\omega )) \\,dP(\\omega )\\\\[6pt]
\n&= P((X \\in B_1 ) \\cap (Y \\in B_2 ))
\n\\end{align}\\]
\n\uc774\uba70, \ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc744 \uacc4\uc0b0\ud558\uba74
\n\\[P(X\\in B_1 ) P(Y\\in B_2 )\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(X\\)\uc640 \\(Y\\)\ub294 \ub3c5\ub9bd\uc774\ub2e4.<\/p>\n
\n\\[\\varphi = \\sum_i b_i \\mathbf{1}_{B_i} ,\\, \\psi = \\sum_j c_j \\mathbf{1} _{C_j}\\]
\n\ub97c (1)\uc5d0 \ub300\uc785\ud558\uace0 \uc120\ud615\uc131\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\mathbb{E} (\\varphi (X) \\psi (Y)) &= \\mathbb{E} \\left( \\sum_i b_i \\mathbf{1}_{B_i} (X) \\sum_j c_j \\mathbf{1} _{C_j} (Y) \\right) \\\\[6pt]
\n&= \\sum_{i,\\,j} b_i c_j \\mathbb{E} (\\mathbf{1}_{B_i} (X) \\mathbf{1}_{C_j} (Y))\\\\[6pt]
\n&= \\sum_{i,\\,j} b_i c_j \\mathbb{E} (\\mathbf{1}_{B_i} (X)) \\mathbb{E} ( \\mathbf{1}_{C_j} (Y))\\\\[6pt]
\n&= \\sum_{i} b_i \\mathbb{E} (\\mathbf{1}_{B_i} (X)) \\sum_{j} c_j \\mathbb{E} ( \\mathbf{1}_{C_j} (Y))\\\\[6pt]
\n&= \\mathbb{E} (\\varphi (X)) \\mathbb{E} (\\psi (Y)).
\n\\end{align}\\]
\n\ud568\uc218 \\(f,\\) \\(g\\)\ub294 \ub2e8\uc21c\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uadfc\uc0ac\uc2dc\ud0ac \uc218 \uc788\uc73c\uba70, \\(f\\)\uc640 \\(g\\)\uac00 \uc720\uacc4\uc774\ubbc0\ub85c \uc9c0\ubc30\uc218\ub834 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74 \uc704 \ub4f1\uc2dd\uc744 \\(f\\)\uc640 \\(g\\)\uc5d0 \ub300\ud55c \ub4f1\uc2dd\uc73c\ub85c \ud655\uc7a5\uc2dc\ud0ac \uc218 \uc788\ub2e4.
\n<\/span><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"