分布函数的概率密度函数计算过程 分布函数与概率密度函数的转化
\u6982\u7387\u5bc6\u5ea6\uff0c\u5206\u5e03\u51fd\u6570\u5206\u5e03\u51fd\u6570\u8f6c\u5316\u4e3a\u6982\u7387\u5bc6\u5ea6\uff0c\u53ea\u9700\u8981\u5bf9\u5206\u5e03\u51fd\u6570\u8fdb\u884c\u6c42\u5bfc\u5c31\u53ef\u4ee5\u6c42\u51fa\u6982\u7387\u5bc6\u5ea6\u3002
\u5982\u679c\u6982\u7387\u5bc6\u5ea6\u4e3a\u8fde\u7eed\u578b\u7684\u6982\u7387\u5bc6\u5ea6\uff0c\u90a3\u4e48\u6c42\u5206\u5e03\u51fd\u6570\u76f4\u63a5\u5bf9\u6982\u7387\u5bc6\u5ea6\u76f4\u63a5\u6c42\u79ef\u5206\u5c31\u53ef\u4ee5\u5f97\u5230\u76f8\u5e94\u7684\u5206\u5e03\u51fd\u6570\u3002
\u5982\u679c\u6982\u7387\u5bc6\u5ea6\u662f\u5206\u6bb5\u51fd\u6570\uff0c\u90a3\u4e48\u6211\u4eec\u5c31\u8981\u4ece\u5206\u5e03\u51fd\u6570\u7684\u5b9a\u4e49\u51fa\u53d1\uff0c\u6765\u6c42\u5206\u5e03\u51fd\u6570\u3002
\u6ce8\u610f\u5206\u5e03\u51fd\u6570\u662f\u7d2f\u52a0\u51fd\u6570\u3002\u5bf9\u6982\u7387\u8fdb\u884c\u9010\u6bb5\u7d2f\u52a0\u5c31\u53ef\u4ee5\u5f97\u5230\u5206\u5e03\u542b\u7a0e\u3002
\u6240\u4ee5\u672c\u9898\u7684\u6982\u7387\u5bc6\u5ea6:
x<0\u65f6 F(x)=\u222b(--\u221e, x)f(x)dx=0\uff0c
\u5f530<=x<1\uff0cF(x)=\u222b(o , x)tdt=\uff08x^2\uff09/2
\u5f531<=x<2,F\uff08x\uff09=\u222b(o , 1)tdt+\u222b\uff081\uff0cx\uff092-tdt=2x-\uff08x^2\uff09/2-1\u3002
\u5f53x>=2\u65f6F\uff08x\uff09=1\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u5206\u5e03\u51fd\u6570\u7684\u6027\u8d28\uff1a
F(x)\u4e3a\u968f\u673a\u53d8\u91cfX\u7684\u5206\u5e03\u51fd\u6570\uff0c\u5176\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u4e3a\uff1a
1\u3001\u975e\u964d\u6027\uff0cF(x)\u662f\u4e00\u4e2a\u4e0d\u51cf\u51fd\u6570
2\u3001\u6709\u754c\u6027
3\u3001\u53f3\u8fde\u7eed\u6027
\u8bbeF\uff08x\uff0cy\uff09 \u662f\u968f\u673a\u53d8\u91cf\uff08x\uff0cy\uff09 \u7684\u5206\u5e03\u51fd\u6570,
1\u30010<=F\uff08x\uff0cy\uff09<=1\u3002
2\u3001\u56fa\u5b9a\u4e00\u4e2a\u81ea\u53d8\u91cf\u7684\u503c\u65f6\uff0c\u4f5c\u4e3a\u4e00\u5143\u51fd\u6570\u5173\u4e8e\u53e6\u4e00\u4e2a\u81ea\u53d8\u91cf\u662f\u5355\u8c03\u4e0d\u51cf\u7684\uff1b
3\u3001\u56fa\u5b9a\u4e00\u4e2a\u81ea\u53d8\u91cf\u7684\u503c\u65f6\uff0cF\uff08x\uff0cy\uff09\u4f5c\u4e3a\u4e00\u5143\u51fd\u6570\u5173\u4e8e\u53e6\u4e00\u4e2a\u81ea\u53d8\u91cf\u81f3\u5c11\u6709\u8fde\u7eed\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u5206\u5e03\u51fd\u6570
p Y= 1=2/3 ,
pY=-1=1/3
绛旓細桅(x)瀹氫箟涓烘湇浠庢爣鍑嗘鎬佸垎甯冪殑闅忔満鍙橀噺X鐨鍒嗗竷鍑芥暟锛屽叾鍊间负瀵筬(x)鍏充簬x绉垎锛屼粠-鈭炵Н鍒皒銆備粠f(x)鍥惧儚涓婄湅锛屛(x)鐨勫肩浉褰撲簬f(x)鏇茬嚎涓涓嬶紝x杞存洸绾夸互涓婏紝鍖哄煙涓猴紙-鈭烇紝x锛夎繖娈电殑闈㈢Н銆傜敱浜巉(x)涓哄伓鍑芥暟锛屼笖鏈夊垎甯冨嚱鏁版ц川桅(+鈭)=1锛屽彲浠ユ眰鍑何(0)=0.5銆傛鎬佸垎甯姒傜巼瀵嗗害鍑芥暟鐗规...
绛旓細绠楀嚭骞冲潎鍊煎拰鏍囧噯宸笺佅冿紝浠ｅ叆姝ｆ鍒嗗竷瀵嗗害鍑芥暟琛ㄨ揪寮忥細f(x) = exp{-(x-渭)²/2蟽²}/[鈭(2蟺)蟽]缁欏畾x鍊,鍗冲彲绠楀嚭f鍊笺
绛旓細宸茬粡鍛婅瘔鍧囧寑鍒嗗竷浜嗭紝鍧囧寑鍒嗗竷鐨勬鐜囧瘑搴﹀嚱鏁灏辨槸1/鍖洪棿闀垮害
绛旓細鎶-a甯﹁繘鍘 -Aa+B=0 锛屾妸a甯﹁繘鍘 Aa+B=1 锛岃仈绔嬭В鍑 AB缁橣姹傚灏卞緱鍑虹殑鏄瀵嗗害鍑芥暟锛乫=a 瀹氫箟鍩熷拰涓婇潰鏄搴旂殑锛
绛旓細娉ㄦ剰鍒嗗竷鍑芥暟鏄疮鍔犲嚱鏁般傚姒傜巼杩涜閫愭绱姞灏卞彲浠ュ緱鍒板垎甯冨惈绋庛傛墍浠ユ湰棰鐨勬鐜囧瘑搴:x<0鏃 F(x)=鈭(--鈭, x)f(x)dx=0锛屽綋0<=x<1锛孎(x)=鈭(o , x)tdt=锛坸^2锛/2 褰1<=x<2,F锛坸锛=鈭(o , 1)tdt+鈭紙1锛寈锛2-tdt=2x-锛坸^2锛/2-1銆傚綋x>=2鏃禙锛坸锛=1銆
绛旓細鍒嗗竷鍑芥暟鐨鍏紡鏄 F锛坸锛=P锛圶<= x锛夎繖涓殑璇濆疄闄呴棶棰樺疄闄呭垎鏋愮殑锛屼竴鑸兘鏄眰鍧囧寑鍒嗗竷鐨鍒嗗竷鍑芥暟锛屾瘮濡傛煇涓闅忔満鍙橀噺鍦0鍒2蟺鐨勬鐜鍧囧寑鍒嗗竷锛岄偅涔堝畠鐨勫垎甯冨嚱鏁板氨鏄疐(X)=X/2蟺.鑰姒傜巼瀵嗗害鍑芥暟灏辨槸瀵瑰垎甯冨嚱鏁扮殑姹傚 銆
绛旓細瀵逛簬浜岀淮杩炵画鍙橀噺鐨鍒嗗竷鍑芥暟F(x锛寉)锛屼竴鑸簲鐢ㄥ叾姒傜巼瀵嗗害鍑芥暟f(x锛寉)鐨勫畾绉垎姹傝В锛涘浜庨潪杩炵画鍙橀噺锛岄渶瑕佸垎鍒疮鍔犳眰寰椼愪笌涓缁撮殢鏈哄彉閲忕殑姹傛硶鐩镐豢銆戙傗埓鏈涓紝褰搙鈭(0,鈭)銆亂鈭(0锛屸垶)鏃讹紝鍒嗗竷鍑芥暟F(x锛寉)=鈭(-鈭烇紝x)du鈭(-鈭,y)f(u锛寁)dv=鈭(0锛寈)du鈭(-0锛寉)2e^(-2u-v...
绛旓細1锛廥锛嶏紙X锛嶈〃绀哄潎鍊硷級銆傝缁嗘眰瑙杩囩▼濡備笅鍥撅細鎸囨暟鍒嗗竷鍙互鐢ㄦ潵琛ㄧず鐙珛闅忔満浜嬩欢鍙戠敓鐨勬椂闂撮棿闅旓紝姣斿鏃呭杩涙満鍦虹殑鏃堕棿闂撮殧銆佷腑鏂囩淮鍩虹櫨绉戞柊鏉＄洰鍑虹幇鐨勬椂闂撮棿闅旂瓑绛夈傛寚鏁板垎甯冨彲浠ョ湅浣滃綋濞佸竷灏斿垎甯冧腑鐨勫舰鐘剁郴鏁扮瓑浜1鐨勭壒娈婂垎甯冿紝鎸囨暟鍒嗗竷鐨勫け鏁堢巼鏄笌鏃堕棿t鏃犲叧鐨勫父鏁帮紝鎵浠鍒嗗竷鍑芥暟绠鍗曘
绛旓細宸茬煡姒傜巼瀵嗗害f(x)锛岄偅涔堟眰F(x)瀵筬(x)杩涜绉垎鍗冲彲锛屽湪x<a鏃讹紝f(x)閮界瓑浜0锛屾樉鐒剁Н鍒咶(x)=0 鑰屽湪a<x
绛旓細瀹為檯璁＄畻鏃堕氳繃鏌ヨ〃鎵炬鐜囥傞潪鏍囧噯姝ｆ鍒嗗竷鍑芥暟鍙互杞崲鎴愭爣鍑嗘鎬佸垎甯冨啀绠椼傝嫢闅忔満鍙橀噺X鏈嶄粠涓涓暟瀛︽湡鏈涗负渭銆佹柟宸负蟽^2鐨勬鎬佸垎甯冿紝璁颁负N(渭锛屜僞2)銆傚叾姒傜巼瀵嗗害鍑芥暟涓烘鎬佸垎甯冪殑鏈熸湜鍊嘉煎喅瀹氫簡鍏朵綅缃紝鍏舵爣鍑嗗樊蟽鍐冲畾浜嗗垎甯冪殑骞呭害銆傚綋渭 = 0,蟽 = 1鏃剁殑姝ｆ佸垎甯冩槸鏍囧噯姝ｆ佸垎甯冦
