设二维随机变量（X，Y）在区域D上服从均匀分布，其中D={（x，y）：|x|+|y|≤1}，又设Z=X+Y．试求（Ⅰ）X 设二维随机变量（X,Y)在区域D上服从均匀分布，其中D:0<...
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\u90a3\u4f60\u5c31\u5728Y=u-X \u548cY=-1-X \u4e4b\u95f4\u5b9a\u79ef\u5206\u533a\u95f4\uff0c(\u4ee5\u7b2c\u4e00\u4e2a\u4e3a\u4f8b)\u6709\u70b9\u9ebb\u70e6
\u7528\u51e0\u4f55\u610f\u4e49\u591a\u7b80\u5355\uff0c\u4f60\u90a3\u6837\u592a\u9ebb\u70e6\u4e86
\u521a\u624d\u628au\u5f04\u9519\u4e86\uff0c\u6211\u76f4\u63a5\u5f53\u6210\u662f\u4e0a\u534a\u90e8\u5206\u4e86\uff0c\u4e0d\u597d\u610f\u601d
D\u533a\u57df\u662f\u4e00\u4e2a\u6b63\u65b9\u5f62
U \u7684\u6982\u7387\u5206\u5e03\uff1a
\u8bbeX+Y\u2264u,\u5219Y\u2264u-X,\u628au\u770b\u6210\u5e38\u6570\uff0c\u5b83\u5c31\u662f\u5728Y\u8f74\u4e0a\u7684\u622a\u8ddd\uff0c\u6700\u5927\u4e3a1\uff0c\u6700\u5c0f\u4e3a-1\uff0c\u6240\u4ee5
F\uff08U) \u5f53U>=1,F(U)=1
\u5f53-1<U<1,F(U)=1-(1-u)/2=(u+1)/2
\u5f53U<=1 F(U)=0,\u6240\u4ee5\u5728\u76f8\u5e94\u533a\u95f4\u5185\u6c42\u5bfc\uff0c\u5f97
f(U)=1/2(-1<u<1),\u5176\u4f59\u4e3a0
\u540c\u7406\u8bbeX-Y\u2264v,\u5219Y>=X-v,-v\u662f\u622a\u8ddd\uff0c\u6700\u540e\u6c42\u5f97
f(V)=1/2(-1<v<1),\u5176\u4f59\u4e3a0
\u4f60\u5e94\u8be5\u5148\u6c42\u51fa\u6982\u7387\u5206\u5e03\uff0c\u6839\u636e\u51e0\u4f55\u610f\u4e49
\u4f60\u90a3\u6837\u4e5f\u53ef\u4ee5\uff0c\u4e24\u6b21\u6c42\u5bfc\uff0c\u7b2c\u4e00\u6b21\uff1a\u222b1/2dx
\u518d\u6c42\u4e00\u6b211/2
\u5f53-1\uff1cx\uff1c0\uff0cf\uff08x\uff09\u221ef\uff08x\uff0cy\uff09dy=2\u222bˣ⁺¹₀\uff081/2\uff09dy=x+1
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D\uff1a0<=x<=2\uff0c0<=y<=2\u662f\u8fb9\u957f\u4e3a2\u7684\u6b63\u65b9\u5f62\u533a\u57df\uff0c\u6240\u4ee5D\u7684\u9762\u79ef\u4e3a4\uff0c\u6545\u6982\u7387\u5bc6\u5ea6\u4e3af\uff08x\uff0cy\uff09=1/4\uff0c\uff08x\uff0cy\uff09\u2208D0\uff0c\u5176\u5b83\u53c8\u56e0\u4e3a\u70b9\uff081\uff0c1\uff09\u5728\u533a\u57dfD\u5185\uff0c\u6240\u4ee5f\uff081\uff0c1\uff09=1/4\u3002
\u6982\u7387\u5bc6\u5ea6
\u5355\u7eaf\u7684\u8bb2\u6982\u7387\u5bc6\u5ea6\u6ca1\u6709\u5b9e\u9645\u7684\u610f\u4e49\uff0c\u5b83\u5fc5\u987b\u6709\u786e\u5b9a\u7684\u6709\u754c\u533a\u95f4\u4e3a\u524d\u63d0\u3002\u53ef\u4ee5\u628a\u6982\u7387\u5bc6\u5ea6\u770b\u6210\u662f\u7eb5\u5750\u6807\uff0c\u533a\u95f4\u770b\u6210\u662f\u6a2a\u5750\u6807\uff0c\u6982\u7387\u5bc6\u5ea6\u5bf9\u533a\u95f4\u7684\u79ef\u5206\u5c31\u662f\u9762\u79ef\uff0c\u800c\u8fd9\u4e2a\u9762\u79ef\u5c31\u662f\u4e8b\u4ef6\u5728\u8fd9\u4e2a\u533a\u95f4\u53d1\u751f\u7684\u6982\u7387\uff0c\u6240\u6709\u9762\u79ef\u7684\u548c\u4e3a1\u3002\u6240\u4ee5\u5355\u72ec\u5206\u6790\u4e00\u4e2a\u70b9\u7684\u6982\u7387\u5bc6\u5ea6\u662f\u6ca1\u6709\u4efb\u4f55\u610f\u4e49\u7684\uff0c\u5b83\u5fc5\u987b\u8981\u6709\u533a\u95f4\u4f5c\u4e3a\u53c2\u8003\u548c\u5bf9\u6bd4\u3002
\u4ee5\u4e0a\u5185\u5bb9\u53c2\u8003\uff1a\u767e\u5ea6\u767e\u79d1-\u6982\u7387\u5bc6\u5ea6
二维随机变量（X，Y）的联合概率密度f(x，y)＝
绛旓細璇烽噰绾 绛旓細闅忔満鍙橀噺(X,Y)鍦ㄥ尯鍩D 鏈嶄粠鍧囧寑鍒嗗竷,鍒欒仈鍚堝瘑搴﹀嚱鏁癙(X,Y)=1/惟,惟=1/2鍗冲尯鍩烡鐨勯潰绉,涓虹洿绾縳=0,y=x,y=1鎵鍥寸殑閮ㄥ垎,鎵浠(X,Y)=2,10,鍥犲叾涓哄潎鍖鍒嗗竷锛屽垯P(X,Y)涓轰竴甯告暟銆傚張P(X,Y)婊¤冻锛氣埆鈭玃(X,Y)dXdY=1锛岃P(X,Y)=k 锛屽垯鏈夆埆鈭玃(X,Y)dXdY=k鈭埆1dXdY=1 锛屸埆... 绛旓細f(x,y) = 1/2, x>0, y>0, x+y 绛旓細瑙ｏ細鍖哄煙闈㈢НS=鈭埆dxdy=4/3 f锛坸,y锛=1/s=3/4,0鈮鈮1,y^2鈮,鍏朵粬涓0 锛2锛塮(x)=鈭 [-鈭,鈭瀅f锛坸,y锛塪y=3鈭歺/2, 0鈮鈮1,鍏朵粬涓0 f(y)=鈭玔-鈭,鈭瀅f(x,y)dx=(3-3y²)/4 锛 -1<=y<=1,鍏朵粬涓0 f(x)f(y)鈮爁(x,y)鏁呬笉鐙珛 涓嶆槑鐧... 绛旓細璁句簩缁撮殢鏈哄彉閲(x,y)鍦骞抽潰鍖哄煙G=锕(x,y)|0<=x<=2,0<=y<=1涓婃湇浠庡潎鍖鍒嗗竷,姹俍=XY鐨勬鐜囧瘑搴︺傞夯鐑﹁缁 鎴戞潵绛 1涓洖绛 #鐑# 娴峰叧鏈夊摢浜涚杩愬晢鍝?鏌ュ埌鍚庢庝箞鍔?fengzhejun77 2014-12-18 路 TA鑾峰緱瓒呰繃3539涓禐 鐭ラ亾澶ф湁鍙负绛斾富 鍥炵瓟閲:989 閲囩撼鐜:75% 甯姪鐨勪汉:1074涓 鎴... 绛旓細cov(X,Y)= -1/36,蟻XY= -1/2,涓嬮潰鏄繃绋嬨(1)鍦ㄤ笁瑙掑舰鍐,鍥犱负瀵嗗害鍧囧寑,鎵浠ユ鐜囧瘑搴﹀嚱鏁皃(X,Y)=1/璇ヤ笁瑙掑舰闈㈢Н=1/(1/2)=2銆(2)璁＄畻E(X),E(X^2),E(Y),E(Y^2),D(X)鍜孌(Y)銆侲(X)=绉垎X(0鍒1) 绉垎Y(0鍒1-x) x*p(x,y) dydx=绉垎X(0鍒1) x(1-x)*2 dx=(x^2-2... 绛旓細P(X<0.5锛孻<0.5)=(0.5脳0.5)/[(1脳1)/2]=0,5 绛旓細锛1锛夊綋-1锛渪锛0锛宖锛坸锛夆垶f锛坸锛寉锛dy=2鈭ˣ⁺¹₀锛1/2锛塪y=x+1 鍚岀悊褰0锛渪锛1锛宖锛坸锛=-x+1锛屾墍浠锛坸锛=x+1锛-1锛渪锛0锛沠锛坸锛=-x+1锛0锛渪锛1锛沠锛坸锛=0銆傚悓鐞嗭細f锛坹锛=y+1锛-1锛測锛0锛沠锛坹锛=-y+1锛0锛測锛1锛沠锛坹锛=0銆... 绛旓細璁緁xy(x,y)=k鍏堟眰k銆傜Н鍒唊涓婁笅闄(0,2-2x/3)dy, (0,3)dx=1=3k, k=1/3 fxy(x,y)=1/3, F(x,y)=xy/3 本站交流只代表网友个人观点，与本站立场无关 |
