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+R2→R2,−R1+R3→R3,−3R1+R3→R3→[111110000110−11000123−101000−2−2−3001]−R2+R3→R3,R4/−2→[111110000110−110000130−11000113/200−1/2]−R2+R1→R1,−R3+R4→R4→[10012−1000110−110000130−110000−23/21−1−1/2]−R3+R2→R2,R4/2→[10012−100010−3−12−1000130−1100001−3/4−1/21/21/4]3R4+R2→R2,−3R4+R3→R3→[10012−1000100−13/41/21/23/400109/41/2−1/2−3/40001−3/4−1/21/21/4]−R4+R1→R1→[100011/4−1/2−1/2−1/40100−13/41/21/23/400109/41/2−1/2−3/40001−3/4−1/21/21/4]⇒A−1=[11/4−1/2−1/2−1/4−13/41/21/23/49/41/2−1/2−3/4−3/4−1/21/21/4]解答:A=[2341]=[1√2−351√245][500−2][1√2−351√245]−1⇒A4=[1√2−351√245][5400(−2)4][1√2−351√245]−1=[625√2−485625√2645][1√2−351√245]−1=[625√2−485625√2645][4√273√27−5757]=[364261348277]因此[0.40.60.80.2]4=(15A)4=1625A4=1625[364261348277]=[364625261625348625277625]
解答:det(A)=4a2−10a=2a(2a−5)=0⇒a=0,52
解答:令{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|=(1√5,2√5,0,0)u2=a2−(a2⋅e1)e1=(1,−3,0,0)+√5(1√5,2√5,0,0)=(2,−1,0,0)⇒e2=u2|u2|=(2√5,−1√5,0,0)u3=a3−(a3⋅e1)e1−(a3⋅e2)e2=(−1,3,1,3)−√5(1√5,2√5,0,0)+√5(2√5,−1√5,0,0)=(0,0,1,3)⇒e3=u3|u3|=(0,0,1√10,3√10)u4=a4−(a4⋅e1)e1−(a4⋅e2)e2−(a4⋅e3)e3=(1,−3,2,−4)+√5(1√5,2√5,0,0)−√5(2√5,−1√5,0,0)+√10(0,0,1√10,3√10)=(0,0,3,−1)⇒e4=u4|u4|=(0,0,3√10,−1√10)因此Q=[e1|e2|e3|e4]⇒Q=[1/√52/√5002/√5−1/√500001/√103/√10003/√10−1/√10]⇒R=[a1⋅e1a2⋅e1a3⋅e1a4⋅e10a2⋅e2a3⋅e2a4⋅e200a3⋅e3a4⋅e3000a4⋅e4]⇒R=[√5−√5√5−√50√5−√5√500√10−√10000√10]
解答:{det(AB)=det(A)det(B)det(AT)=det(A)⇒det(AB)=|003−42−33−3001211−11||12−12−343−60014003−8|=−50×(−200)=10000
解答:(a)fX(x)=∫∞010e−(5x+2y)dy=[−5e−(5x+2y)]|∞0=5e−5x⇒fX(x)=5e−5x,x≥0(b)E(X)=∫∞05xe−5xdx=[−15e−5x(5x+1)]|∞0=15(c)fY(y)=∫∞010e−(5x+2y)dx=2e−2y⇒FY(x)=∫x02e−2ydy=1−e−2x同理,FX(x)=∫x05e−5xdx=1−e−5x因此CDF of min(X,Y)=P(min(X,Y)≤z)=FX(z)+FY(z)−FX(z)FY(z)=(1−e−5z)+(1−e−2z)−(1−e−5z)(1−e−2z)=1−e−7z⇒CDF of min(X,Y)=1−e−7z,z=min(X,Y)
解答:(a)PX(x=1)+PX(x=2)+PX(x=3)=1⇒c+c2+c3=11c6=1⇒c=611(b)P[2≤X≤8]=P(X=2)+P(X=3)=c2+c3=5c6=56⋅611=511(c)EX2=12P(x=1)+22P(X=2)+32P(X=3)=1⋅c1+4⋅c2+9⋅c3=6c=3611
解答:(a)No:{E1 and E2 are disjoint ⇒P(E1∩E2)=P(∅)=0E1 and E2 are independent ⇒P(E1∩E2)=P(E1)P(E2),兩者不一定相等(b)No:P(E1∪E2)=P(E1)+P(E2)−P(E1∩E2)(c)No:連續型機率密度函數沒有單點機率值(d)No:∫∞0e−2xdx=12≠1
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