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2023年10月9日 星期一

110年台北大學碩士班-工程數學詳解

國立臺北大學110學年度碩士班一般入學考試

系(所)組別:通訊工程系
科目:工程數學(線性代數與機率)

解答[11111000122101001234001033110001]R1+R2R2,R1+R3R3,3R1+R3R3[11111000011011000123101000223001]R2+R3R3,R4/2[11111000011011000013011000113/2001/2]R2+R1R1,R3+R4R4[10012100011011000013011000023/2111/2]R3+R2R2,R4/2[10012100010312100013011000013/41/21/21/4]3R4+R2R2,3R4+R3R3[10012100010013/41/21/23/400109/41/21/23/400013/41/21/21/4]R4+R1R1[100011/41/21/21/4010013/41/21/23/400109/41/21/23/400013/41/21/21/4]A1=[11/41/21/21/413/41/21/23/49/41/21/23/43/41/21/21/4]
解答A=[2341]=[12351245][5002][12351245]1A4=[12351245][5400(2)4][12351245]1=[62524856252645][12351245]1=[62524856252645][4273275757]=[364261348277][0.40.60.80.2]4=(15A)4=1625A4=1625[364261348277]=[364625261625348625277625]
解答det(A)=4a210a=2a(2a5)=0a=0,52
解答{a1=(1,2,0,0)a2=(1,3,0,0)a3=(1,3,1,3)a4=(1,3,2,4)u1=a1=(1,2,0,0)e1=u1|u1|=(15,25,0,0)u2=a2(a2e1)e1=(1,3,0,0)+5(15,25,0,0)=(2,1,0,0)e2=u2|u2|=(25,15,0,0)u3=a3(a3e1)e1(a3e2)e2=(1,3,1,3)5(15,25,0,0)+5(25,15,0,0)=(0,0,1,3)e3=u3|u3|=(0,0,110,310)u4=a4(a4e1)e1(a4e2)e2(a4e3)e3=(1,3,2,4)+5(15,25,0,0)5(25,15,0,0)+10(0,0,110,310)=(0,0,3,1)e4=u4|u4|=(0,0,310,110)Q=[e1|e2|e3|e4]Q=[1/52/5002/51/500001/103/10003/101/10]R=[a1e1a2e1a3e1a4e10a2e2a3e2a4e200a3e3a4e3000a4e4]R=[5555055500101000010]
解答{det(AB)=det(A)det(B)det(AT)=det(A)det(AB)=|0034233300121111||1212343600140038|=50×(200)=10000
解答(a)fX(x)=010e(5x+2y)dy=[5e(5x+2y)]|0=5e5xfX(x)=5e5x,x0(b)E(X)=05xe5xdx=[15e5x(5x+1)]|0=15(c)fY(y)=010e(5x+2y)dx=2e2yFY(x)=x02e2ydy=1e2xFX(x)=x05e5xdx=1e5xCDF of min(X,Y)=P(min(X,Y)z)=FX(z)+FY(z)FX(z)FY(z)=(1e5z)+(1e2z)(1e5z)(1e2z)=1e7zCDF of min(X,Y)=1e7z,z=min(X,Y)
解答(a)PX(x=1)+PX(x=2)+PX(x=3)=1c+c2+c3=11c6=1c=611(b)P[2X8]=P(X=2)+P(X=3)=c2+c3=5c6=56611=511(c)EX2=12P(x=1)+22P(X=2)+32P(X=3)=1c1+4c2+9c3=6c=3611
解答(a)No:{E1 and E2 are disjoint P(E1E2)=P()=0E1 and E2 are independent P(E1E2)=P(E1)P(E2)(b)No:P(E1E2)=P(E1)+P(E2)P(E1E2)(c)No:(d)No:0e2xdx=121

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