Assumption: X1 , X2 , X3 are independent
a.
P(0.2<X1<1 , 0.2<X2<1.5 ,
0.25<X3<0.8) = P(0.2<X2<1.5)
P(0.25<X3<0.8)
P(0.2<X1<1 , 0.2<X2<1.5 , 0.25<X3<0.8) = (e-0.8-e-4)(e-0.8-e-6)(e-1-e-3.2) = e-2.6-e-7.8-e-5.8+e-11-e-4.8+e-10+e-8-e-13.2 = 0.063
b.
E[2560X1(X2-0.25)2(X3-0.25)2]
= 2560E(X1X22X32-5X1X22X3+0.0625X1X22-5X1X2X32+25X1X2X3-0.3125X1X2+0.0625X1X32-0.3125X1X3-0.00390625)
= 2560[
E(X1X22X32)
- 5E(X1X22X3) +
0.0625E(X1X22) -
5E(X1X2X32) +
25E(X1X2X3) -
0.3125E(X1X2) +
0.0625E(X1X32) -
0.3125E(X1X3) - 0.00390625 ]
= 2560[
E(X1)E(X22)E(X32)
- 5E(X1)E(X22)E(X3) +
0.0625E(X1)E(X22) -
5E(X1)E(X2)E(X32) +
25E(X1)E(X2)E(X3) -
0.3125E(X1)E(X2) +
0.0625E(X1)E(X32) -
0.3125E(X1)E(X3) - 0.00390625 ]
E[2560X1(X2-0.25)2(X3-0.25)2]
= 2560[ 0.25
0.125
0.125
- 5
0.25
0.125
0.25 +
0.0625
0.25
0.125
- 5
0.25
0.25
0.125
+ 25
0.25
0.25
0.25 -
0.3125
0.25
0.25 +
0.0625
0.25
0.125
- 0.3125
0.25
0.25 -
0.00390625 ]
= 25600.27734375 =
710
3 from the exponential distribu- Let X1,ng and tion with pdf be a randon sample of size n f(x) -4e-4x, 0 < x < oo. Find a. P(0.2< X1,0.2< X2 < 1.5,0.25< X3< 0.8) b. E[2560X1 (X2-0.25)"(Xy-0.25判·
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how to calculate cov(x1,x2), cov(x2,x3),cov(x3,x1)?
and how to calculate var(x1),var(x2),var(x3)?
Given three random variables Xi, X2, and X such that X[Xi X2 X 20 -1 E [X] ,1-10 | and var(X)=Σ-| 0 3 0. 1 0.5 1 compuite: 2