设二维随机变量(X,Y)的概率密度函数为 Z=2X-Y的概率密度函数 设X和Y是两个相互独立的随机变量,其概率密度分别为,求随机变...
\u8bbe\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf\uff08X\uff0cY\uff09\u7684\u6982\u7387\u5bc6\u5ea6\u4e3af\uff08x\uff0cy\uff09=1\uff0c0\uff1cx\uff1c1\uff0c0\uff1cy\uff1c2x0\uff0c\u5176\u5b83\uff0c\u6c42Z=2X-Y\u7684\u6982\u7387\u5bc6\u5ea6Z=2X-Y\u7684\u5206\u5e03\u51fd\u6570\u4e3a\uff1aFZ\uff08z\uff09=P{Z\u2264z}\uff0c\u5f53z\u22640\u65f6\uff0cFZ\uff08z\uff09=0\uff0c\u5f530\uff1cz\uff1c1\u65f6\uff0cz=2x-y\u4e0ex\u8f74\u7684\u4ea4\u70b9\u4e3a\uff1a(z2\uff0c0)\uff0c\u4e0ex=1\u7684\u4ea4\u70b9\u4e3a\uff081\uff0c2-z\uff09\uff0cFZ\uff08z\uff09=1-12\u00d7(1?z2)\u00d7(2?z)=1-(1?z2)2\uff0c\u5f53z\u22651\u65f6\uff0cFZ\uff08z\uff09=1\uff0c\u6545FZ\uff08z\uff09=0\uff0c z\u226401?(1?z2)2\uff0c 0\uff1cz\uff1c11\uff0c z\u22651\uff0e\u6c42\u5bfc\u53ef\u5f97\uff0cZ=2X-Y\u7684\u5bc6\u5ea6\u51fd\u6570\u4e3a\uff1afZ\uff08z\uff09=FZ\uff08z\uff09=2(1?z2)\uff0c 0\uff1cz\uff1c10\uff0c \u5176\u4ed6\uff0e
x=0~1\uff0cy=0~+\u221e\uff0cz=0~2x+y\uff08\u5e73\u9762\uff09\u7684\u4e00\u4e2a\u534a\u65e0\u9650\u7acb\u4f53\uff0c\u662f\u6982\u7387\u7a7a\u95f4\u3002
z=2X+Y
Y=-2X+z\uff0cX=0\uff0cY=z\uff1bY=0\uff0cX=z/2\uff1b
\uff081\uff09X=z/2\u22641\uff0cz\u22642\uff1b
P\uff08Z\u2264z\uff09=\u222b\uff080\uff0cz/2\uff09fx\uff08x\uff09dx\u222b\uff080\uff0c-2x+z\uff09fy\uff08y\uff09dy
=\u222b\uff080\uff0cz/2\uff091dx\u222b\uff080\uff0c-2x+z\uff09e^(-y)dy
=-\u222b\uff080\uff0cz/2\uff09[e^(-y)]\uff080\uff0c-2x+z\uff09dx
=-\u222b\uff080\uff0cz/2\uff09[e^(2x-z)-1]dx
=-[0.5e^(2x-z)-x]\uff080\uff0cz/2\uff09
=-[0.5-z/2-0.5e^(-z)]
fz(z)=-[-1/2+0.5e^(-z)]=1/2-0.5/e^z=0.5(1-1/e^z)
z\uff1e2
P\uff08Z\u2264z\uff09=\u222b\uff080,1\uff09fx(x)dx\u222b\uff080\uff0c-2x+z\uff09fy\uff08y\uff09dy
=\u222b\uff080,1\uff09dx\u222b\uff080\uff0c-2x+z\uff09e^(-y)dy
=-\u222b\uff080,1\uff09[e^(-y)]\uff080\uff0c-2x+z\uff09dx
=-\u222b\uff080,1\uff09[e^(2x-z)-1]dx
=-[0.5e^(2x-z)-x]\uff080,1\uff09
=-[0.5e^(2-z)-1-0.5e^(-z)]
=1+0.5e^(-z)-0.5e^(2-z)
\u6c42\u5bfc
fz(z)=-0.5e^(-z)+0.5e^(2-z)
=0.5e^(-z)[e²-1]
\u6269\u5c55\u8d44\u6599\uff1a
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\u8fde\u7eed\u578b\u7684\u968f\u673a\u53d8\u91cf\u53d6\u503c\u5728\u4efb\u610f\u4e00\u70b9\u7684\u6982\u7387\u90fd\u662f0\u3002\u4f5c\u4e3a\u63a8\u8bba\uff0c\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u5728\u533a\u95f4\u4e0a\u53d6\u503c\u7684\u6982\u7387\u4e0e\u8fd9\u4e2a\u533a\u95f4\u662f\u5f00\u533a\u95f4\u8fd8\u662f\u95ed\u533a\u95f4\u65e0\u5173\u3002\u8981\u6ce8\u610f\u7684\u662f\uff0c\u6982\u7387P{x=a}=0\uff0c\u4f46{X=a}\u5e76\u4e0d\u662f\u4e0d\u53ef\u80fd\u4e8b\u4ef6\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u6982\u7387\u5bc6\u5ea6\u51fd\u6570
简单计算一下即可，答案如图所示
见图
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