Homework Answers
Var(Z) = Var(2X + 3Y) = 4*Var(X) + 9*Var(Y) + 2*2*3*Cov(X, Y)
= 4*Var(X) + 9*Var(Y) + 2*2*3*Corr(X, Y)*SD(X)*SD(Y) ----- (1)
[SD means standard deviation. Note that Corr(X, Y) = Cov(X, Y)/SD(X)SD(Y)]
Given that Var(Y) = 4*Var(X). Hence, 
Replacing this in above eqution (1), we get Var(Z) = 4*Var(X) +
9*4*Var(X) + 2*2*3*(1/3)*SD(X)*2*SD(X)
Or, 240 = 40*Var(X) + 8*SD(X)*SD(X) = 40*Var(X) + 8*Var(X) = 48*Var(X)
Hence, Var(X) = 240/48 = 5
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Suppose Cor(X,Y)=1/3 and oy = 403. Let Z=2X +3Y. If Var(Z) = 240, calculate Var(X).
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Suppose XX and YY are independent random variables for which Var(X)=7Var(X)=7 and Var(Y)=7.Var(Y)=7. (a) Find Var(X−Y+1).Var(X−Y+1). (b) Find Var(2X−3Y)Var(2X−3Y) (c) Let W=2X−3Y.W=2X−3Y. Find the standard deviaton of W.W.
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3. Suppose x,y,z satisfy the competing species equations <(6 - 2x – 3y - 2) y(7 - 2x - 3y - 22) z(5 - 2x - y -22) (a) (6 points) Find the critical point (0,Ye, ze) where ye, we >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable. (c) (8 points) Determine if the critical point (2,0,0) is stable, where I > 0.
For random variables X, Y, and Z, Var(X) = 4, Var(Y) = 9, Var(Z) = 16,...
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Obtain E(Z|X), Var(Z|X) and verify that E(E(Z|X)) =E(Z), Var(E(Z|X))+E(Var(Z|X)) =Var(Z) 3. Let X, Y be independent...
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3. Suppose x, y, z satisfy the competing species equations 2(6 - 2x - 3y -...
3. Suppose x, y, z satisfy the competing species equations 2(6 - 2x - 3y - 2) y(7 - 2.0 - 3y - 22) z(5 – 2x - y -22) (a) (6 points) Find the critical point (0, yc, ze) where yc, ze >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable. (c) (8 points) Determine if the critical point (1,0,0) is stable, where 8c > 0.
3. Suppose x, y, z satisfy the competing species equations 2(6 - 2x - 3y -...
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Let X, Y be independent random variables with E[X] = E[Y] = 0 and ox =...
Let X, Y be independent random variables with E[X] = E[Y] = 0 and ox = Oy = 5. Then Var(2x+3Y) = 1. True False
3. Suppose x,y,z satisfy the competing species equations <(6 - 2x – 3y - 2) y(7 - 2x - 3y - 22) z(5 - 2x - y -22) (a) (6 points) Find the critical point (0,Ye, ze) where ye, we >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable. (c) (8 points) Determine if the critical point (2,0,0) is stable, where I > 0.
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Obtain E(Z|X), Var(Z|X) and verify that E(E(Z|X)) =E(Z),
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3. Let X, Y be independent Exponential (1) random variables. Define 1, if X Y<2 Obtain E (Z|X), Var(ZX) and verify that E(E(Zx)) E(Z), Var(E(Z|X))+E(Var(Z|X)) - Var(Z)
(4, -1,2) 2x + 3y - z = 3 x + y - z = 5 10x – 2y = 3 SHOP Supa lo color (4, 1) ) boda e 3x - 5y = 7 2x + 2y = 10
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3. Suppose x, y, z satisfy the competing species equations 2(6 - 2x - 3y - 2) y(7 - 2.0 - 3y - 22) z(5 – 2x - y -22) (a) (6 points) Find the critical point (0, yc, ze) where yc, ze >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable. (c) (8 points) Determine if the critical point (1,0,0) is stable, where 8c > 0.
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Let X, Y be independent random variables with E[X] = E[Y] = 0 and ox = Oy = 5. Then Var(2x+3Y) = 1. True False
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